HOMOTOPY EQUIVALENCES OF p-COMPLETED CLASSIFYING SPACES OF FINITE GROUPS

CARLES BROTO, RAN LEVI, AND BOB OLIVER

Abstract. We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category L_{p}^{c}(G) with the following properties. Two p-completed classifying spaces BG_{p}^{∧} and BG′_{p}^{∧} have the same homotopy type if and only if the associated categories L_{p}^{c}(G) and L_{p}^{c}(G′) are equivalent. And the topological group Aut(BG_{p}^{∧}) of self equivalences is determined by the self equivalences of the associated category L_{p}^{c}(G).

Throughout this paper, let p denote a fixed prime. For any finite group G, we write BG_{p}^{∧} to denote the Bousfield-Kan p-completion of BG. The main results in this paper are algebraic conditions for the existence of homotopy equivalences between BG_{p}^{∧} and BG′_{p}^{∧}, for any pair of finite groups G and G′, and a description of the space of self homotopy equivalences of BG_{p}^{∧}.

Spaces of the form BG_{p}^{∧} for G finite are not generally aspherical. In fact, if G contains a p-perfect subgroup N (i.e. H_{1}(BN) has order prime to p), whose order is divisible by p, then BG_{p}^{∧} has non-vanishing higher homotopy groups in arbitrarily high dimensions. For a general discussion of BG_{p}^{∧} for G finite we refer to [Lev].

This work originated as a study by the first two authors of the space of self equivalences of BG_{p}^{∧}, which was motivated by the problem of classifying fibrations with BG_{p}^{∧} as fiber. Some of their work on this subject appears in [BL]. A possibly more general motivation for the project is the analogy with compact Lie groups. For compact connected simple Lie groups G the space of self maps of BG was studied by Jackowski, McClure and Oliver [JMO]. In their study homology decompositions for classifying spaces were used to provide a description of the corresponding mapping spaces.

Homology decompositions are one of the most important tools in the study of classifying spaces and maps between them. Our approach however is different. Central to our study are certain categories L_{p}^{c}(G), defined for any finite group G, which we proceed by describing.

1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 55Q52.

Key words and phrases. Classifying space, p-completion, finite groups.

C. Broto is partially supported by DGICYT grant PB97–0203.

R. Levi is partially supported by EPSRC grant GR/M7831.

B. Oliver is partially supported by UMR 7539 of the CNRS.

2 Homotopy equivalences of p-completed classifying spaces

Recall that a p-subgroup P ≤ G is p-centric if its center Z(P) is the Sylow psubgroup of its centralizer C_{G}(P). Equivalently, the centralizer of such a group naturally splits as a cartesian product C_{G}(P) ≅ Z(P) × C′_{G}(P), where C′_{G}(P) is a group of order prime to p. Objects of L_{p}^{c}(G) are p-centric subgroups P ≤ G. For any pair of such subgroups P, Q ≤ G,

Mor_{L_{p}^{c}(G)}(P, Q) = N_{G}(P, Q)/C′_{P} (G),

where N_{G}(P, Q) = {x ∈ G | xP x^{−1} ≤ Q}.

Our first theorem is the following:

Theorem A. For any prime p and a pair G, G′ of finite groups, the p-completed classifying spaces BG_{p}^{∧} and BG′_{p}^{∧} are homotopy equivalent if and only if the categories L_{p}^{c}(G) and L_{p}^{c}(G′) are equivalent.

The work in [BL] was motivated in part by the paper [MP], where the authors claim an algebraic condition for determining whether or not two distinct p-completed classifying spaces have the same homotopy type. In the course of trying to compare this claim to their own work, the authors of this article were led to the discovery of an error in [MP], which invalidates the proof of the main theorem there. We still do not know whether or not their theorem is true, but our attempts to fix the error led to Theorem A, which gives a different algebraic criterion for the existence of an equivalence.

We next consider the topological monoid Aut(BG_{p}^{∧}) of all self homotopy equivalences of BG_{p}^{∧}. Let Out(BG_{p}^{∧}) denote the group of components π_{0}(Aut(BG_{p}^{∧})). For any category C, let Aut(C) denote the strict monoidal category whose objects are self equivalences F : C → C and whose morphisms are natural isomorphisms of functors. Notice that every morphism in Aut(C) is invertible making it into a groupoid. Let Out(C) denotes the group of components of Aut(C), namely the set of equivalence classes obtained by identifying two functors F′, F′′ ∈ Aut(C) if they are naturally isomorphic. These uses of the notation Out(−) are motivated by the observation that Out(BG) ≅ Out(B(G)) ≅ Out(G) for any discrete group G (where B(G) is the category with one object and morphism group G).

The categories L_{p}^{c}(G) come equipped with a natural forgetful functor λ_{G} from L_{p}^{c}(G) to the category of groups. A self equivalence ψ of the category L_{p}^{c}(G) is called isotypical if the functor λ_{G} and the composite functor λ_{G}ψ are naturally isomorphic. The full subcategory Aut_{typ}(L_{p}^{c}(G)) of Aut(L_{p}^{c}(G)) whose objects are all isotypical self equivalences of L_{p}^{c}(G) inherits a monoidal structure. We let Out_{typ}(L_{p}^{c}(G)) ≤ Out(L_{p}^{c}(G)) denote the subgroup of classes of isotypical self equivalences. Our next theorem gives a description of Out(BG_{p}^{∧}).

Theorem B. For any prime p and a finite group G, there is an isomorphism of groups

Out_{typ}(L_{p}^{c}(G)) ⟶^{≅} Out(BG_{p}^{∧}).

The homotopy groups of a small category C can be defined purely algebraically or equivalently as the homotopy groups of the geometric realization of its nerve. Theorem B claims in fact that Out(BG_{p}^{∧}) = π_{0}(Aut(BG_{p}^{∧})) is isomorphic as a group to π_{0}(Aut_{typ}(L_{p}^{c}(G))). The next theorem generalizes Theorem B by showing that in fact the groupoid Aut_{typ}(L_{p}^{c}(G)) completely describes the homotopy type of Aut(BG_{p}^{∧}). The proof makes use of the fact, proven in [BL], that Aut(BG_{p}^{∧}) is aspherical. For a small category C let |C| denote the geometric realization of the nerve of C.

Theorem C. For any prime p and a finite group G there is a weak homotopy equivalence

Aut(BG_{p}^{∧}) ≃ |Aut_{typ}(L_{p}^{c}(G))|.

In fact these spaces are weakly equivalent as topological monoids, in the sense that their classifying spaces are also weakly homotopy equivalent.

Theorem C is stated in terms of weak equivalences rather than homotopy equivalence because of the usual problem of whether or not a mapping space between two CW complexes has itself the homotopy type of a CW complex. If one is willing to work simplicially, i.e. replace spaces by their singular simplicial sets and mapping spaces by the respective simplicial mapping spaces, then one may replace “weak equivalence” by “equivalence” in the statement of the theorem. Throughout the paper we shall not distinguish between homotopy equivalences and weak homotopy equivalences, but the above comment should be kept in mind.

The identity component of Aut(BG_{p}^{∧}) was shown in [BL] to have the homotopy type of BZ(Ḡ), where Ḡ = G/O_{p′}G, the quotient of G by its maximal normal subgroup of order prime to p.

Theorem A will be proven in a more precise form as Theorem 2.9 below. Theorems B and C will be proven together as Theorem 4.5.

Let G and G′ be finite groups, with Sylow p-subgroups S ≤ G and S′ ≤ G′. An isomorphism φ : S ⟶^{≅} S′ is called fusion preserving if for any P, Q ≤ S, a group isomorphism α : P ^{≅}→ Q is induced by conjugation in G if and only if the corresponding isomorphism φαφ^{−1} : φ(P) → φ(Q) is induced by conjugation in G′.

The claim of the main theorem in [MP] can be reformulated in those terms. However an error in its proof occurs on [MP, p.129], where a certain functor fails to be well defined on morphisms. As we have neither a proof nor a counterexample of this statement we restate it as the following

Conjecture D (Martino-Priddy [MP]). For any prime p and a pair G, G′ of finite groups, BG_{p}^{∧} ≃ BG′_{p}^{∧} if and only if there is a fusion preserving isomorphism between their Sylow p-subgroups.

Theorem A gives another purely algebraic condition for the spaces BG_{p}^{∧} and BG′_{p}^{∧} to be homotopy equivalent. However, the condition in the Martino-Priddy Conjecture is simpler to check in practice. It is easier in general to construct an isomorphism between Sylow subgroups and show that it preserves fusion, than to construct an equivalence between categories L_{p}^{c}(−).

In Section 6, we study the relation between Conjecture D and our Theorem A. Let Z be the contravariant functor on the orbit category for p-centric subgroups of G which sends P to its center Z(P). Let G be a finite group with a Sylow p-subgroup S. Let Out_{fus}(S) denote the quotient of the group of fusion preserving automorphisms of S by the normal subgroup of all automorphisms induced by conjugation in G. The following theorem is an algebraic analogue of [BL, Thm. 1.6].

Theorem E. For any prime p and a finite group G, there is an exact sequence

0 ⟶ \underleftarrow{lim}^{1}_{O_{p}^{c}(G)}(Z) ⟶^{λ_{G}} Out_{typ}(L_{p}^{c}(G)) ⟶^{μ_{G}} Out_{fus}(S) ⟶^{ω_{G}} \underleftarrow{lim}^{2}_{O_{p}^{c}(G)}(Z),

where λ_{G} and μ_{G} are group homomorphisms.

The \underleftarrow{lim}^{1} term in the theorem is known not to vanish in general. The \underleftarrow{lim}^{2} term is the group containing the obstructions for lifting fusion preserving outer automorphisms of S to isotypical outer automorphisms of L_{p}^{c}(G). We know of no group G for which the \underleftarrow{lim}^{2} obstruction group is non-vanishing (much less the obstruction itself not vanishing). The best we can show towards proving Conjecture D is the following (see Theorems 5.10 and 6.5 below):

Theorem F. For any prime p and a pair of finite groups G and G′, Conjecture D is true if either of the following two conditions on G holds:

(i) rk_{p}(G) < p^{2}, or

(ii) there is no chain P ≨ Q ≨ S, where S ≤ G is a Sylow p-subgroup, and P and Q are both p-centric and p-radical in G.

Here, a p-subgroup P ≤ G is p-radical, or p-stubborn, if N(P)/P contains no nontrivial normal p-subgroups.

The paper is organized as follows. In Section 1, we define the categories L_{p}^{c}(G) and discuss some of their basic properties. In Section 2 we construct the categories L_{p}^{c}(X) for an arbitrary space X and prove Theorem A. Sections 3 and 4 are devoted to the proof of Theorems B and C. In section 5, we discuss higher limits obstructions to the existence and uniqueness of a lift of a given fusion preserving isomorphism between Sylow subgroups to a homotopy equivalence between the respective classifying spaces and prove Theorem E and the first half of Theorem F. Section 6 is concerned with the construction of self equivalences of the categories L_{p}^{c}(G) by direct methods (i.e. without using higher limits). Finally in Section 7, we give a number of examples to demonstrate the use of our methods.

The authors would like to express their gratitude to Hans-Werner Henn, John Martino, Stewart Priddy and Peter Webb for many useful conversations and exchanges of e-mail. We would particularly like to thank Bill Dwyer and Jesper Grodal who have expressed a strong interest in this work since its early stages. We have greatly benefited from talking to them.

1. From groups to categories

Fix a finite group G and a prime p. We begin by defining the categories L_{p}(G) and L_{p}(G) which will play a central role throughout this paper. An object in L_{p}(G) is a p-subgroup P ≤ G. For any pair of p-subgroups P, Q ≤ G, we define

Mor_{L_{p}(G)}(P, Q) = N_{G}(P, Q) =^{def} {x ∈ G | xP x^{−1} ≤ Q}.

The morphism from P to Q corresponding to x ∈ N_{G}(P, Q) will be denoted x̂. Composition is given by x̂ ∘ ŷ = x̂y.

As usual, for any finite group H, O^{p}(H) denotes the maximal normal p-perfect subgroup of H; equivalently the minimal normal subgroup of p-power index. We define L_{p}(G) to be the category with the same objects as L_{p}(G), and with morphism sets

Mor_{L_{p}(G)}(P, Q) = N_{G}(P, Q)/O^{p}(C_{G}(P)).

In other words, we have divided out by the action of O^{p}(C_{G}(P)) on N_{G}(P, Q) given by right multiplication. In particular, L_{p}(G) is a quotient category of L_{p}(G). The morphism P → Q corresponding to the class of x ∈ N_{G}(P, Q) will also be denoted x̂.

Our goal here is to construct a category, depending on G and p, which reflects the structure of BG_{p}^{∧} in a way which will be made clear later. In order to do this, we must further restrict the categories L_{p}(G) and L_{p}(G). Following the terminology of Dwyer, we call a subgroup H ≤ G centric if C_{G}(H) ≤ H (i.e., if C_{G}(H) = Z(H)). A p-subgroup P ≤ G is p-centric if its centralizer has the form

C_{G}(P) = Z(P) × C′_{G}(P),

where C′_{G}(P) has order prime to p. Equivalently, P is p-centric if and only if |O^{p}(C_{G}(P))| is prime to p. We let L_{p}^{c}(G) ⊆ L_{p}(G) and L_{p}^{c}(G) ⊆ L_{p}(G) denote the full subcategories whose objects are the p-centric subgroups of G. In particular, for any p-centric P, Q ≤ G,

Mor_{L_{p}^{c}(G)}(P, Q) = N_{G}(P, Q)/C′_{G}(P).

For any finite group G, let B(G) denote the category with one object o_{G} and morphism group G. The morphism in B(G) corresponding to x ∈ G will be denoted x̃, and x̃ ∘ ỹ = x̃y. We identify the classifying space BG with the (realization of the) nerve of B(G). Let

α_{G}: |L_{p}(G)| ⟶ |B(G)| = BG

be the nerve of the functor L_{p}(G) → B(G) which sends each object of L_{p}(G) to o_{G}, and which sends a morphism P \underrightarrow{x̂} Q (for x ∈ N_{G}(P, Q)) to x̃ ∈ Aut_{B(G)}(o_{G}). The main result of this section is the following:

Proposition 1.1. For any finite group G, the maps

α_{G}: |L_{p}^{c}(G)| ⟶ BG  and  |π|: |L_{p}^{c}(G)| ⟶ |L_{p}^{c}(G)|

are both 𝔽_{p}-homology equivalences. Hence there is a homotopy equivalence

α_{G}: |L_{p}^{c}(G)|_{p}^{∧} ⟶^{≃} BG_{p}^{∧},

unique up to homotopy, such that the following triangle of homotopy equivalences commutes up to homotopy:

\xymatrix{|L_{p}^{c}(G)|_{p}^{∧} \ar[rr]^{|π|_{p}^{∧}}_{≃} \ar[dr]_{(α_{G})_{p}^{∧}}^{≃} &  & |L_{p}^{c}(G)|_{p}^{∧} \ar[dl]^{_{α_{G}}}_{≃} \\  & BG_{p}^{∧}.}

Proof. This follows from Lemmas 1.2 and 1.3 below. Lemma 1.2 says that α_{G} is an 𝔽_{p}-homology equivalence, and hence induces a homotopy equivalence (α_{G})_{p}^{∧} after p-completion. Lemma 1.3 implies as a special case that the projection functor π: L_{p}^{c}(G) → L_{p}^{c}(G) induces an 𝔽_{p}-homology equivalence between the nerves, and hence a homotopy equivalence between their p-completions. The other statements are then clear.

The rest of the section is devoted to the proofs of Lemmas 1.2 and 1.3. Recall [Dw] that a family C of p-subgroups of G (closed under conjugation) is called ample if the projection EG ×_{G} |K_{C}| → BG is an 𝔽_{p}-homology equivalence. Here, |K_{C}| denotes the realization of the nerve of the poset C, and the action of G is induced by conjugation. In particular, the family of p-centric subgroups of G is ample by [Dw, §8]. In general we will denote by L_{p}^{C}(G) ⊆ L_{p}(G) and L_{p}^{C}(G) ⊆ L_{p}(G) the full subcategories whose objects are in C.

Lemma 1.2. For any finite group G, any prime p, and any ample family C of psubgroups of G, α_{G} restricts to an 𝔽_{p}-homology equivalence

α_{G}^{C}: |L_{p}^{C}(G)| ⟶ BG.

Proof. Let K_{C} denote the category whose set of objects is C, and with a unique morphism P → Q whenever P ≤ Q. Let E(G) be the category with objects the elements of G and with a unique morphism between each pair of objects (i.e., the category whose nerve is the universal free G-space EG). Let

γ: E(G) ×_{G} K_{C} ⟶ L_{p}^{C}(G)

be the functor which sends a pair of objects (g, P) to ^{g}P = gP g^{−1}, and which sends a pair of morphisms (g → xg, P ≤Q) to x̂: ^{g}P → ^{xg}Q. This clearly induces bijections on objects and on morphisms, and hence a homeomorphism

|γ|: EG ×_{G} |K_{C}| ⟶^{≅} |L_{p}^{C}(G)|.

Let δ: E(G) ×_{G} K_{C} → B(G) be the functor which sends all objects to o_{G}, and which sends a morphism (g → xg, P ≤Q) to x̃ ∈ Aut_{B(G)}(o_{G}). The following triangle of categories and functors

\xymatrix{E(G) ×_{G} K_{C} \ar[rr]^{γ} \ar[dr]_{δ} &  & L_{p}^{C}(G) \ar@{-->}[dl]^{α_{G}^{C}} \\  & B(G)}

commutes by definition, and hence induces a commutative triangle of spaces

\xymatrix{EG ×_{G} |K_{C}| \ar[rr]^{|γ|}_{≅} \ar[dr]_{|δ|} &  & |L_{p}^{C}(G)| \ar[dl]^{|α_{G}^{C}|} \\  & BG.}

Since |δ| is an 𝔽_{p}-homology equivalence (by definition of “ample”), |α_{G}^{C}| is also an 𝔽_{p}-homology equivalence.

The following lemma will be used to show that the projection |L_{p}^{c}(G)| → |L_{p}^{c}(G)| is an 𝔽_{p}-homology equivalence.

Lemma 1.3. Fix a prime p. Let ψ: C ⟶ C′ be any functor between small categories which has the following properties:

(i) ψ is bijective on isomorphism classes of objects and is surjective on morphism sets.

(ii) For each object c in C, the subgroup

K(c) =^{def} Ker[Aut_{C}(c) ⟶ Aut_{C′}(ψ(c))]

is finite of order prime to p.

(iii) For each pair of objects c and d, and each f, g: c → d in C, ψ(f) = ψ(g) if and only if there is some σ ∈ K(c) such that g = f ∘ σ (i.e., Mor_{C′}(ψ(c), ψ(d)) ≅ Mor_{C}(c, d)/K(c)).

Then for any functor F: C′ → Spaces , the induced map

\underrightarrow{hocolim}_{C}(F) ⟶ \underrightarrow{hocolim}_{C′}(F ∘ ψ)

is an 𝔽_{p}-homology equivalence, and hence induces a homotopy equivalence between the p-completions. Also, for any functor T: C′^{op} → ℤ_{(p)}-mod,

\underleftarrow{lim}_{C}^{∗}(T) ≅ \underleftarrow{lim}_{C′}^{∗}(T ∘ ψ).

Proof. We can assume that ψ is bijective on objects (not just on isomorphism classes). If not, then replace C′ by the equivalent category C′′ whose objects are those of C, and where Mor_{C′′}(c_{1}, c_{2}) = Mor_{C′}(ψ(c_{1}), ψ(c_{2})).

Let C-mod and C′-mod denote the categories of functors C^{op} → Ab and C′^{op} → Ab. We proceed by showing that the functor ψ^{∗} : C′-mod → C-mod given by composition on the right with ψ has a left adjoint, which is a useful observation in proving our claim. Define (−)_{K} : C-mod → C′-mod by

D_{K}(c′) = D(c)_{K(c)} =^{def} ℤ ⊗_{K(c)} D(c) = H_{0}(K(c); D(c))

for any D in C-mod, any c′ ∈ C′ and c = ψ^{−1}(c′). Note that c is uniquely defined by the assumption that ψ is bijective on objects. To see that D_{K} is well defined as a (contravariant) functor on C′, let f′ : ψ(c) → ψ(d) be any morphism and let f: c → d be a lift to C. Then for any τ ∈ K(d) there exists, by (iii) above (applied with g = τ ∘ f), some σ ∈ K(c) such that f ∘ σ = τ ∘ f, and any other lift of f′ differs from f by an element of K(c).

By construction, for any D in C-mod and D′ in C′-mod,

Hom_{C′-mod} (D_{K}, D′) ≅ Hom_{C-mod} (D, D′ ∘ ψ),  (1)

and we have thus shown that (−)_{K} is left adjoint to ψ^{∗}. In particular, this shows that D_{K} is C′-mod-projective if D is C-mod-projective. So if

⋯ ⟶ P_{2} ⟶ P_{1} ⟶ P_{0} ⟶ \underline{ℤ} ⟶ 0

is any C-mod projective resolution of the constant functor ℤ, then

⋯ ⟶ (P_{2})_{K} ⟶ (P_{1})_{K} ⟶ (P_{0})_{K} ⟶ \underline{ℤ} ⟶ 0

is a chain complex of C′-mod-projectives which is exact (a resolution) after localization at p (the groups K(c) all being finite of order prime to p). Since

\underleftarrow{lim}_{C}(T ∘ ψ) ≅ Hom_{C-mod} (\underline{ℤ}, T ∘ ψ),

we have

\underleftarrow{lim}_{C}^{∗}(T ∘ ψ) ≅ Ext_{C-mod}^{∗} (\underline{ℤ}, T ∘ ψ) ≅ H^{∗}(Hom_{C-mod} (P_{∗}, T ∘ ψ)).

Similarly, since T (c) is a ℤ_{(p)}-module for all c in C′,

\underleftarrow{lim}_{C′}^{∗}(T) ≅ Ext_{C′-mod}^{∗} (\underline{ℤ}, T) ≅ H^{∗}(Hom_{C-mod} ((P_{∗})_{K}, T)).

So \underleftarrow{lim}_{C}^{∗}(T ∘ ψ) ≅ \underleftarrow{lim}_{C′}^{∗}(T) by (1).

The 𝔽_{p}-homology equivalence between homotopy colimits now follows from the isomorphisms

\underleftarrow{lim}_{C}^{∗} H^{∗}(F ∘ ψ(−); 𝔽_{p}) ≅ \underleftarrow{lim}_{C′}^{∗} H^{∗}(F (−); 𝔽_{p}),

together with the spectral sequence for the cohomology of a homotopy colimit.

Remark 1.4. According to [Dw], any ample family C of p-subgroups of a finite group G gives rise to an 𝔽_{p}-homology equivalence

\underrightarrow{hocolim}_{P_{p}^{C}(G)} EG/C_{G}(−) ⟶ BG

where P_{p}^{C}(G) is the Frobenius category whose objects are the subgroups in C and whose morphisms are homomorphisms induced by conjugation in G. That is,

Mor_{P_{p}^{C}(G)}(P, Q) ≅ N_{G}(P, Q)/C_{G}(P) .

By the same methods, we can obtain another 𝔽_{p}-homology equivalence

\underrightarrow{hocolim}_{L_{p}^{C}(G)} EG/O^{p}(C_{G}(−)) → BG.

The relationship between these two decompositions can be described as moving information from the functor in the first decomposition to the indexing category in the second. The case corresponding to the discussion above is the family of subgroups P such that O^{p}(C_{G}(P)) has order prime to p, which holds if and only if P is p-centric. In this case EG/O^{p}(C_{G}(P)) ≃ BO^{p}(C_{G}(P)) is p-acyclic, which explains the existence of the homotopy equivalence α_{G}: |L_{p}^{c}(G)|_{p}^{∧} ^{≃}→ BG_{p}^{∧} of Proposition 1.1 in terms of the above 𝔽_{p}-homology equivalences, and also why we do not expect a similar equivalence to be true for families of subgroups which are not p-centric.

2. From spaces to discrete categories

In this section we introduce a homotopy theoretic analogue of the categories L_{p}(G): a category L_{p}(X), defined for an arbitrary space X. The remarkable feature of this construction is that L_{p}(BG) ≃ L_{p}(G) and L_{p}(BG_{p}^{∧}) ≃ L_{p}(G). The following proposition suggests a way of doing this. For any pair of groups G, H, we set

Rep(G, H) = Hom(G, H)/ Inn(H) :

the set of homomorphisms G → H modulo conjugacy in H.

Proposition 2.1. For any finite p-group P and any finite group G, the p-completion map BG → BG_{p}^{∧} induces a (weak) homotopy equivalence

Map(BP, BG)_{p}^{∧} ⟶^{≃} Map(BP, BG_{p}^{∧}).

In particular, the maps

Rep(P, G) ⟶_{≅}^{B} [BP, BG] ⟶_{≅}^{κ_{∗}} [BP, BG_{p}^{∧}],

where B sends a homomorphism ρ: P → G to Bρ, are bijections. Also, for each ρ: P → G, the induced product map C_{G}(ρ) × P → G induces (after taking adjoints) a homotopy equivalence

BC_{G}(ρ) ⟶ Map(BP, BG)_{Bρ}

and a weak homotopy equivalence

BC_{G}(ρ)_{p}^{∧} ⟶ Map(BP, BG_{p}^{∧})_{Bρ} .

Proof. The description of Map(BP, BG) is classical, and in fact holds for any pair of discrete groups. The fact that Map(BP, BG_{p}^{∧}) is the p-completion of Map(BP, BG) (when P is a p-group and G is finite) is shown in [BL, Proposition 2.1].

Proposition 2.1 suggests replacing a homomorphism P → G, for a p-group P, by a map BP → X. We next state a homotopy theoretic condition which corresponds to a homomorphism being injective. There could be several natural choices for such a condition. The following definition of a homotopy monomorphism seems to be the most useful for our purposes here.

Definition 2.2. A map f: X → Y is said to be a homotopy monomorphism at p if H^{∗}(X; 𝔽_{p}) is a finitely generated module over H^{∗}(Y; 𝔽_{p}) via the induced map f^{∗}. For any space X and any prime p, a subgroup of X at p (a p-subgroup of X) is a pair (G, j) where G is a group (a p-group) and j: BG → X is a homotopy monomorphism.

This definition of homotopy monomorphisms at p is motivated by the following lemma.

Lemma 2.3. Let G and G′ be two finite groups and f: G → G′ a homomorphism. Then Bf: BG → BG′_{p}^{∧} is a homotopy monomorphism at p if and only if Ker(f) has order prime to p.

Proof. This statement is implicit in [Sw]. Assume first that f: G → G′ is a monomorphism, and fix an embedding G′ ⊆ U(n) for some n. The composite G → G′ → U(n) induces a fibration

U(n)/G ⟶ BG ⟶ BU(n),

and the E^{2}-term of the spectral sequence for this fibration is finitely generated as a module over H^{∗}(BU(n); 𝔽_{p}). Hence, since H^{∗}(BU(n); 𝔽_{p}) is noetherian, the E^{∞}- term is also finitely generated. The composite BG → BG′ → BU(n) thus makes H^{∗}(G; 𝔽_{p}) into a finitely generated H^{∗}(BU(n); 𝔽_{p})-module; and hence a finitely generated H^{∗}(G′; 𝔽_{p})-module.

If f: G → G′ is such that K = Ker(f) has order prime to p, then H^{∗}(G; 𝔽_{p}) ≅ H^{∗}(G/K; 𝔽_{p}) by the Serre spectral sequence for the fibration BK → BG → B(G/K). So H^{∗}(G; 𝔽_{p}) is finitely generated as an H^{∗}(G′; 𝔽_{p})-module in this case also.

Conversely, assume that H^{∗}(G; 𝔽_{p}) is finitely generated over H^{∗}(G′; 𝔽_{p}), and again set K = Ker(f). Since K ≤ G, H^{∗}(K; 𝔽_{p}) is a finitely generated H^{∗}(G; 𝔽_{p})-module, and thus a finitely generated H^{∗}(G′; 𝔽_{p})-module. Since f|_{K}: K → G′ is trivial, Im[H^{∗}(G′; 𝔽_{p}) → H^{∗}(K; 𝔽_{p})] = 𝔽_{p}; and hence dim_{𝔽_{p}}(H^{∗}(K; 𝔽_{p})) < ∞. And by [Sw, Corollary 1], this implies that |K| is prime to p.

We also need to define the analogue of “p-centric subgroups” of a space. Recall (Proposition 2.1) that for any finite group G and any p-subgroup P ≤ G, BC_{G}(P) ≃ Map(BP, BG)_{incl} and BC_{G}(P)_{p}^{∧} ≃ Map(BP, BG_{p}^{∧})_{incl}. This motivates the following definition, originally due to Dwyer and Kan [DK].

Definition 2.4. A map f: X → Y is centric if the induced map

(f ∘ −): Map(X, X)_{id} → Map(X, Y)_{f}

is a homotopy equivalence, and is p-centric if (f ∘ −) is an 𝔽_{p}-homology equivalence. A p-subgroup (P, j) of a space X is p-centric if the map j: BP → X is p-centric.

We are now ready to define the categories L_{p}^{c}(X) ⊆ L_{p}(X).

Definition 2.5. Fix a space X. Define L_{p}(X) to be the category whose objects are the p-subgroups (P, j) of X; and where a morphism from (P, j) to (Q, k) is a pair (φ, η), where φ: P → Q is a homomorphism, and where η represents a homotopy class of paths in Map(BP, X) from j to k ∘ Bφ. Let L_{p}^{c}(X) be the full subcategory of L_{p}(X) whose objects are the p-centric subgroups of X.

If f: X → Y is a homotopy monomorphism at p, then there is an induced functor L_{p}(f): L_{p}(X) → L_{p}(Y), and this correspondence satisfies the usual functorial properties. In order to obtain similar functoriality properties for the construction L_{p}^{c} we need maps to preserve p-centricity; that is, for (P, j) a p-centric subgroup of X we need that (P, f ∘ j) is also p-centric in Y. This happens only under very restrictive conditions. One relevant case of interest to us is that of the p-completion map BG → BG_{p}^{∧}, for a finite group G. This is clearly a homotopy monomorphism at p and by Proposition 2.1, preserves p-centricity. It therefore induces functors

L_{p}(κ): L_{p}(BG) → L_{p}(BG_{p}^{∧}) and L_{p}^{c}(κ): L_{p}^{c}(BG) → L_{p}^{c}(BG_{p}^{∧}) .

For any finite group G, any p-subgroups P, Q ≤ G, and any x ∈ N_{G}(P, Q), we let c_{x}: P → Q denote the homomorphism c_{x}(g) = xgx^{−1}. Also, η_{x}: BP × I → BG denotes the homotopy (i.e., path in Map(BP, BG)) induced by the natural transformation o_{P} ↦ x̃ ∈ Aut_{B(G)}(o_{G}) between the functors B(incl), Bc_{x}: B(P) → B(G).

Proposition 2.6. For any finite group G and any prime p, let

β_{G}: L_{p}(G) ⟶ L_{p}(BG)

be the functor defined as follows. For any p-subgroup P ≤ G, set β_{G}(P) = (P, i_{P}), where i_{P} : BP → BG denotes the inclusion. For any pair of p-subgroups P, Q ≤ G, and any x ∈ N_{G}(P, Q), set

β_{G}(x̂) = (c_{x}, η_{x}): (P, i_{P}) ⟶ (Q, i_{Q}).

Then β_{G} is an equivalence of categories, and restricts to an equivalence

β_{G}^{c}: L_{p}^{c}(G) ⟶ L_{p}^{c}(BG).

Proof. By Proposition 2.1, [BP, BG] ≅ Rep(P, G) for any finite group P. By Lemma 2.3, a homomorphism ρ: P → G is injective if and only if Bρ: BP → BG is a homotopy monomorphism. This shows that β_{G} induces a bijection between isomorphism classes of objects in L_{p}(G) and in L_{p}(BG).

By Proposition 2.1 again, for any p-subgroup P ≤ G, (P, i_{P}) is p-centric in the sense of Definition 2.4 if and only if the inclusion BZ(P) → BC_{G}(P) is an 𝔽_{p}-homology equivalence. Since Z(P) is central in C_{G}(P), the inclusion induces an 𝔽_{p}-homology equivalence if and only if C_{G}(P)/Z(P) has order prime to p, if and only if P is pcentric in G. This shows that β_{G}^{c} also induces a bijection between isomorphism classes of objects.

It remains to show, for all P, Q ≤ G, that β_{G} induces a bijection

Mor_{L_{p}(G)}(P, Q) → Mor_{L_{p}(BG)}((P, i_{P}), (Q, i_{Q})).

For any x ∈ N_{G}(P, Q) (i.e., any morphism x̂: P → Q in L_{p}(G)), β_{G}(x̂) = (c_{x}, η_{x}), and by construction the restriction of η_{x}: BP × I → BG to the basepoint of BP is a loop which represents the element x ∈ π_{1}(G) ≅ G. This proves that the map from Mor_{L_{p}(G)}(P, Q) to Mor_{L_{p}(BG)}((P, i_{P}), (Q, i_{Q})) is injective. So it remains only to show that these morphism sets have the same order.

By definition, Mor_{L_{p}(BG)}((P, i_{P}), (Q, i_{Q})) consists of pairs (ρ, η) where ρ: P → Q is a homomorphism and η represents a homotopy class of paths in Map(BP, BG) from i_{P} to i_{Q} ∘ Bρ. For any fixed ρ, π_{1}(Map(BP, BG)_{i_{P}}) ≅ C_{G}(P) acts freely and transitively on the set of homotopy classes of paths of Map(BP, BG) from i_{P} to i_{Q}∘Bρ. Hence π_{1}(Map(BP, BG)_{i_{P}}) ≅ C_{G}(P) acts freely on Mor_{L_{p}(BG)}((P, i_{P}), (Q, i_{Q})) and orbits are in one to one correspondence with all homomorphism P → Q induced by conjugation; that is, N_{G}(P, Q)/C_{G}(P). In particular,

| Mor_{L_{p}(BG)}((P, i_{P}), (Q, i_{Q}))| = |N_{G}(P, Q)/C_{G}(P)|·|C_{G}(P)| = = |N_{G}(P, Q)| = | Mor_{L_{p}(G)}(P, Q)| .

We next show that the categories L_{p}^{c}(G) and L_{p}^{c}(BG_{p}^{∧}) are equivalent for any finite

group G. This uses the description in the previous section of maps BP → BG_{p}^{∧} for p-groups P.

Proposition 2.7. The composite functor

L_{p}(G) ⟶^{β_{G}} L_{p}(BG) ⟶^{L_{p}(κ)} L_{p}(BG_{p}^{∧})

factors through an equivalence of categories

β_{G}: L_{p}(G) ⟶ L_{p}(BG_{p}^{∧}),

which restricts to an equivalence of subcategories

β_{G}^{c}: L_{p}^{c}(G) ⟶ L_{p}^{c}(BG_{p}^{∧}).

Proof. Define β_{G} on objects by setting β_{G}(P) = (P, i_{P}) for any p-subgroup P ≤ G (where i_{P} now denotes BP → BG_{p}^{∧}, the p-completion of the map induced by inclusion). Since [BP, BG_{p}^{∧}] ≅ [BP, BG] for any p-group P (Proposition 2.1), this induces a bijection between isomorphism classes of objects in L_{p}(G) and those in L_{p}(BG_{p}^{∧}). Also, i_{P} is p-centric as a map to BG_{p}^{∧} if and only if it is p-centric as a map to BG (Proposition 2.1 again), and this holds if and only if P is p-centric by Proposition 2.6.

It remains to show, for any pair of p-subgroups P, Q ≤ G, that there is a (unique) bijection β_{G}(P, Q) which makes the following square commute:

N_{G}(P, Q) = Mor_{L_{p}(G)}(P, Q) \overline{→^{β_{G}(P,Q) ; ≅}} Mor_{L_{p}(BG)}((P, i_{P}), (Q, i_{Q}))

N_{G}(P, Q)/C′_{G}(P) = Mor_{L_{p}(G)}(P, Q) ↓ ↓  →^{β_{G}(P,Q)} Mor_{L_{p}(BG_{p}^{∧})}((P, i_{P}), (Q, i_{Q})). ↓ ↓^{κ(P,Q)}

The same argument as that used in the proof of Proposition 2.6 shows that the group π_{1}(Map(BP, BG_{p}^{∧})_{Bi_{P}}) acts freely on Mor_{L_{p}(BG_{p}^{∧})}((P, i_{P}), (Q, i_{Q})) with orbit set N_{G}(P, Q)/C_{G}(P), the same as the orbit set of the action of π_{1}(Map(BP, BG)_{Bi_{P}}) on Mor_{L_{p}(BG)}((P, i_{P}), (Q, i_{Q})). Furthermore, the map κ(P, Q) is equivariant via the induced map of fundamental groups

π_{1}(Map(BP, BG)_{Bi_{P}}) ⟶ π_{1}(Map(BP, BG_{p}^{∧})_{Bi_{P}}) .  (1)

By Proposition 2.1,

Map(BP, BG_{p}^{∧})_{i_{P}} ≃ BC_{G}(P)_{p}^{∧} ≃ (Map(BP, BG)_{i_{P}} )_{p}^{∧}.

For any finite group L, L/O^{p}(L) is a p-group, hence B(L/O^{p}(L)) is p-complete, BO^{p}(L)_{p}^{∧} is simply connected by [BK, VII.3.2], and the sequence

BO^{p}(L)_{p}^{∧} → BL_{p}^{∧} → B(L/O^{p}(L))

is a fibration sequence by [BK, II.5.2(iv)], hence

π_{1}(BL_{p}^{∧}) ≅ L/O^{p}(L) .

In particular, the homomorphism of equation (1) is onto, hence so is κ(P, Q).

Also, for any x, y ∈ N_{G}(P, Q) such that c_{x} = c_{y} ∈ Hom(P, Q), η_{x} and η_{y} are homotopic as paths in Map(BP, BG_{p}^{∧}) from i_{P} to i_{Q} ∘ Bc_{x} if and only if η_{x}η_{y}^{−1} is trivial in π_{1}(Map(BP, BG_{p}^{∧})_{i_{P}} ); that is, if and only if [xy^{−1}] = 1 ∈ π_{1}(BC_{G}(P)_{p}^{∧}), if and only if xy^{−1} ∈ O^{p}(C_{G}(P)), if and only if x̂ = ŷ in Mor_{L_{p}(G)}(P, Q). And this

shows that β_{G}(P, Q) is well defined and a bijection. Since β_{G} takes p-centric objects to p-centric objects, its restriction to L_{p}^{c}(G) is also an equivalence of categories.

Upon combining Propositions 2.6 and 2.7 with Lemma 1.3, we get the following:

Corollary 2.8. For any finite group G, the functor L_{p}^{c}(κ): L_{p}^{c}(BG) → L_{p}^{c}(BG_{p}^{∧}) induces an 𝔽_{p}-homology equivalence on nerves.

We are now ready to prove Theorem A, which we restate as

Theorem 2.9. For any pair G, G′ of finite groups, BG_{p}^{∧} and BG′_{p}^{∧} are homotopy equivalent if and only if the categories L_{p}^{c}(G) and L_{p}^{c}(G′) are equivalent.

Proof. A homotopy equivalence BG_{p}^{∧} → BG′_{p}^{∧} induces an equivalence of categories L_{p}^{c}(BG_{p}^{∧}) → L_{p}^{c}(BG′_{p}^{∧}), and hence an equivalence L_{p}^{c}(G) ⟶^{≃} L_{p}^{c}(G′) by Proposition 2.7. Conversely, if the categories L_{p}^{c}(G) and L_{p}^{c}(G′) are equivalent, then their nerves are homotopy equivalent, and hence BG_{p}^{∧} ≃ BG′_{p}^{∧} by Proposition 1.1.

3. Spaces of equivalences

We now turn to the proofs of Theorems B and C. They will first be reduced, in Lemma 3.1, to a problem of comparing two “categories of automorphisms”: one of the space BG_{p}^{∧}, and the other of the category L_{p}^{c}(G). We begin by defining these categories of automorphisms.

For any category C, we denote by Aut(C) the category whose objects are the equivalences of categories ψ : C → C, and whose morphisms are the natural isomorphisms Ψ : ψ → ψ′ between equivalences. We think of an equivalence between categories either as a functor which is invertible modulo natural isomorphisms of functors, or equivalently as a functor which induces bijections on isomorphism classes of objects and on all morphism sets. Analogously, for any CW complex X, we denote by Aut(X) the fundamental groupoid of Aut(X); i.e., the category whose objects are the self homotopy equivalences φ : X ^{≃}→ X, and where Mor_{Aut(X)}(φ, φ′) is the set of homotopy classes of paths from φ to φ′ in the mapping space. All of these categories Aut(−) are given the discrete topology. In both cases, we write Out(−) = π_{0}(Aut(−)). Thus, Out(C) is the group of self equivalences of C up to natural isomorphisms of functors, and Out(X) is the group of homotopy classes of self homotopy equivalences of X.

In general, we write C ^{≃}→ D to indicate that a functor is an equivalence of categories, and ψ ≅ ψ′ to denote that two functors ψ and ψ′ are naturally isomorphic. When functors ψ, ψ′: C → D are already defined, a natural isomorphism ψ ^{Φ}→ ψ′ will be described as a function which sends each object c ∈ Ob(C) to an isomorphism Ψ_{c} in D, satisfying the obvious naturality properties.

Recall that a natural morphism ψ ^{Φ}→ ψ′ of functors C → D is equivalent to a functor Φ̂: C × [1] → D, where [1] is the category with to objects 0, 1 and one non-identity morphism (0 → 1). For each c ∈ Ob(C), Φ̂ sends (c, 0) to ψ(c) and (c, 1) to ψ′(c), while for each morphism c ^{f}→ d,

Φ̂(f, Id_{0}) = ψ(f), Φ̂(f, Id_{1}) = ψ′(f), and Φ̂(f, 0 → 1) = Φ_{d} ∘ ψ(f) = ψ′(f) ∘ Φ_{c}.

Each of these categories Aut(−) is a strict monoidal category, in the sense that composition defines a strictly associative functor

Aut(−) × Aut(−) ⟶ Aut(−)

with strict identity. The nerve of each Aut(−) is thus a simplicial monoid, and |Aut(−)| is a topological monoid.

For any space X, let S_{•}X denote its singular simplicial set, and let π(X) denote its fundamental groupoid. Let

X ⟵^{ev_{X}} |S_{•}X| ⟶^{σ_{X}} |π(X)|

denote the evaluation map, and the map which takes each simplex to its homotopy class, respectively. The following lemma shows that Aut(BG_{p}^{∧}) = π(Aut(BG_{p}^{∧})) and Aut(BG_{p}^{∧}) have the same homotopy type, and hence that it suffices from now on to work with the former.

Lemma 3.1. For any finite group G,

Aut(BG_{p}^{∧}) ⟵_{≃}^{ev} |S_{•} Aut(BG_{p}^{∧})| ⟶_{≃}^{σ} |Aut(BG_{p}^{∧})|

are homotopy equivalences of topological monoids.

Proof. Both maps are clearly morphisms of topological monoids. The first is a (weak) homotopy equivalence (cf. [GJ, Theorem I.11.4]), and the second is a homotopy equivalence since by [BL, Theorem 1.1], Aut(BG_{p}^{∧}) is aspherical.

It now remains to compare Aut(BG_{p}^{∧}) with Aut(L_{p}^{c}(G)). But in fact, we do not deal with all self equivalences of the category L_{p}^{c}(G), but only certain “isotypical” equivalences.

Definition 3.2. Let C and D be two categories equipped with functors C ^{γ}→ Gr and D ^{δ}→ Gr to the category of groups. A functor ψ : C → D is isotypical if γ is naturally isomorphic to δ ∘ ψ. When γ is understood, Aut_{typ}(C) denotes the strict monoidal category of isotypical self-equivalences and natural isomorphisms between them, and Out_{typ}(C) denotes the group of isotypical self-equivalences of C modulo natural isomorphisms of functors.

We emphasize that the definition of an isotypical functor does not include the natural isomorphism γ ^{≅}→ δ ∘ ψ as part of the data, but only requires that such an isomorphism exists. Thus, if we think of a category together with a functor to groups as a “diagram of groups”, then an equivalence between diagrams of groups defines an isotypical equivalence, but the isotypical equivalence contains less information than the equivalence between diagrams.

The following lemma is immediate, and shows for example that Out_{typ}(C) is always a group.

Lemma 3.3. Let C and D be categories equipped with functors C ^{γ} → Gr and D ^{δ} → Gr, and let ψ : C → D be an isotypical functor. Then any functor ψ′ : C → D naturally isomorphic to ψ is isotypical, and any right inverse to ψ up to natural isomorphism (i.e., any D ^{ψ^{∗}}→ C such that ψ ∘ ψ^{∗} ≅ Id_{D}) is isotypical.

The categories L_{p}(G), L_{p}(G), L_{p}(X) and all their variations are examples of such diagrams of groups, where the functors

λ_{G} : L_{p}(G) ⟶ Gr and λ_{X} : L_{p}(X) ⟶ Gr,

are the obvious forgetful functors

λ_{G}(P x̂→ Q) = (P ^{c_{x}}→ Q) and λ_{X}((P, α) ^{(φ,η)} → (Q, β)) = (P ^{φ}→ Q).

(Recall that c_{x} is the homomorphism g ↦ xgx^{−1}.)

For the categories L_{p}^{c}(G), there is an alternative criterion for a functor being isotypical, which is more useful in concrete situations. For each p-centric subgroup P ≤ G, let D_{P} (G) denote the “distinguished subgroup” of Aut_{L_{p}^{c}(G)}(P) given by

D_{G}(P) = {ĝ| g ∈ P } ⊆ Aut_{L_{p}^{c}(G)}(P) ≅ N_{G}(P)/C′_{G}(P).

Since C′_{G}(P) has order prime to p, D_{G}(P) ≅ P, and we can identify these two groups. The next lemma says that an equivalence ψ: L_{p}^{c}(G) ^{≃}→ L_{p}^{c}(G′) is isotypical if and only if it sends distinguished subgroups isomorphically to distinguished subgroups.

Lemma 3.4. Fix finite groups G and G′, and an equivalence ψ: L_{p}^{c}(G) → L_{p}^{c}(G′). Then ψ is isotypical if and only if

ψ_{P,P} : Aut_{L_{p}^{c}(G)}(P) ⟶^{≅} Aut_{L_{p}^{c}(G′)}(ψ(P))

sends D_{G}(P) isomorphically to D_{G′}(ψ(P)) for each p-centric P ≤ G. In this case, if P ^{ψ_{P}}→ ψ(P) denotes the restriction of ψ_{P,P} under the identifications P = D_{G}(P) and ψ(P) = D_{G′}(ψ(P)), then (P ↦ ψ_{P}) is a natural isomorphism of functors λ_{G} ^{≅}→ λ_{G′}∘ ψ.

Proof. To simplify notation, we write P′ = ψ(P) for any P in L_{p}^{c}(G). Assume first

that ψ is isotypical, and let Λ: λ_{G} ^{≅}→ λ_{G′} ∘ ψ be a natural isomorphism. Fix P, let g ∈ P be any element, and set x̂ = ψ_{P,P} (ĝ), where x ∈ N_{G′}(P′). Then Λ sends P to an isomorphism Λ_{P} ∈ Iso(P, P′) of groups, and

c_{Λ_{P} (g)} ∘ Λ_{P} = Λ_{P} ∘ c_{g} = c_{x} ∘ Λ_{P} :

the first equality holds when Λ_{P} is replaced by any homomorphism P → P′, and the second holds by the naturality of Λ with respect to (P ĝ→ P). Thus c_{Λ_{P} (g)} = c_{x}, so x^{−1}Λ_{P} (g) ∈ C_{G′}(P′), and we can assume that x^{−1}Λ_{P} (g) ∈ Z(P′) without changing the class x̂ = x·C′_{G′}(P′). It follows that x ∈ P′, i.e. that ψ_{P,P} (ĝ) ∈ D_{G′}(P′), and thus ψ_{P,P} (D_{G}(P)) ⊆ D_{G′}(P′). And equality now holds since the distinguished subgroups are abstractly isomorphic (and ψ_{P,P} is an isomorphism).

Now assume that ψ_{P,P} (D_{G}(P)) = D_{G′}(P′) for each P, and let ψ_{P} : P ^{≅}→ P′ be the restriction of ψ_{P,P} under our identifications. We must show that (P ↦ ψ_{P}) is natural as an isomorphism of functors λ_{G} → λ_{G′} ∘ ψ; i.e., that

ψ_{Q} ∘ c_{x} = c_{y} ∘ ψ_{P} ∈ Hom(P, Q′)  (1)

for any morphism P x̂→ Q in L_{p}^{c}(G), where ŷ = ψ_{P,Q}(x̂). For any g ∈ P, ψ sends x̂ ∘ ĝ = ^{ x̂}g ∘ x̂ in L_{p}^{c}(G) to

ŷ ∘ { \atop ψ_{P} (g)} = { \atop ψ_{Q}(^{x}g)} ∘ ŷ ∈ Mor_{L_{p}^{c}(G′)}(P′, Q′) = N_{G′}(P′, Q′)/C′_{G′}(P′),

where C_{G′}(P′) = Z(P′) × C′_{G′}(P′) and the second factor has order prime to p. Hence there is h ∈ C′_{G′}(P′) such that yψ_{P} (g) = ψ_{Q}(xgx^{−1})yh,

yhy^{−1} = ψ_{Q}(xgx^{−1})^{−1}yψ_{P} (g)y^{−1} ∈ Q′

(since yP′y^{−1} ≤ Q′), and h = 1 since it has order prime to p. Thus, ψ_{Q}(xgx^{−1}) = yψ_{P} (g)y^{−1} for all g ∈ P, and this proves (1).

From now on, for any isotypical equivalence L_{p}^{c}(G) \begin{matrix}ψ \\ ≃ \end{matrix}→ L_{p}^{c}(G′), and any p-centric

P ≤ G, we let ψ_{P} : P ^{≅}→ ψ(P) denote the isomorphism obtained by restricting ψ_{P,P}.

We next define functors

Aut(BG_{p}^{∧}) \overrightarrow{⟵}_{R}^{L} Aut_{typ}(L_{p}^{c}(G)),

which will later be seen to be inverses up to natural isomorphism. Very roughly, R is defined to be the realization functor from Aut_{typ}(L_{p}^{c}(G)) to Aut(|L_{p}^{c}(G)|_{p}^{∧}) followed by an equivalence of categories induced by the homotopy equivalence |L_{p}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}; while L is induced by the functor L(−) to Aut_{typ}(L_{p}^{c}(BG_{p}^{∧})) followed by an equivalence induced by L_{p}^{c}(BG_{p}^{∧}) ≃ L_{p}^{c}(G).

More precisely, for any category C, let

Aut(C) ⟶^{|−|} Aut(|C|) and Aut(C) ⟶^{|−|_{p}^{∧}} Aut(|C|_{p}^{∧})

denote the functors which take a self equivalence of C to its geometric realization (before or after p-completion), and which take a natural isomorphism, interpreted as a functor from C ×[1] to C, to the homotopy class of its geometric realization. For any finite group G, let |L_{p}^{c}(G)|_{p}^{∧} ^{α_{G}}→ BG_{p}^{∧} be the homotopy equivalence of Proposition 1.1, and fix a homotopy inverse

α^{∗} : BG_{p}^{∧} ⟶ |L_{p}^{c}(G)|_{p}^{∧}.

Let

c_{α} : Aut(|L_{p}^{c}(G)|_{p}^{∧}) ⟶ Aut(BG_{p}^{∧})

denote the functor “conjugation by α”, defined on objects and morphisms by setting

c_{α}(f) = α^{∗} ∘ f ∘ α_{G} and c_{α}(F) = α^{∗} ∘ F ∘ (α_{G} × I)

for any f ∈ Aut(|L_{p}^{c}(G)|_{p}^{∧}) and any homotopy F. We now define the realization functor

R = R_{G}: Aut_{typ}(L_{p}^{c}(G)) ⟶^{|−|_{p}^{∧}} Aut(|L_{p}^{c}(G)|_{p}^{∧}) ⟶_{≃}^{c_{α}} Aut(BG_{p}^{∧}).

The first functor in this composite preserves the monoidal structures of the categories, but c_{α} does not in general.

For any space X, define a functor

L_{X} : Aut(X) ⟶ Aut_{typ}(L_{p}^{c}(X))

as follows. On objects, L_{X} sends a self equivalence X ^{f}→ X to L_{p}^{c}(f). If F :

X × I → X is a homotopy, representing a morphism in Aut(X) from f to f′, then L_{X}(F) is defined to be the natural isomorphism of functors which sends an object (P, α) to the morphism (Id_{P} , [F ∘ (α × I)]). (Note that this only depends on the homotopy class of F, as a path in Aut(X) from f to f′.) One easily checks that L_{X} preserves compositions of homotopies and of homotopy equivalences, and is thus a a well defined functor of monoidal categories.

Since β_{G}^{c} : L_{p}^{c}(G) → L_{p}^{c}(BG_{p}^{∧}) is an inclusion and an equivalence of categories (Proposition 2.7), it has a left inverse β^{∗}, defined by sending any object (P, α) in L_{p}^{c}(BG_{p}^{∧}) not in the image of β_{G}^{c} to some Q ≤ G such that (Q, i_{Q}) = β_{G}^{c}(Q) is isomorphic to (P, α) in L_{p}^{c}(BG_{p}^{∧}). Also, β_{G}^{c} is isotypical since λ_{G} = λ_{BG_{p}^{∧}} ∘ β_{G}^{c}, and hence β^{∗} is isotypical by Lemma 3.3. Let

Aut_{typ}(L_{p}^{c}(BG_{p}^{∧})) \overrightarrow{⟵}_{c_{β}^{∗}}^{c_{β}} Aut_{typ}(L_{p}^{c}(G))

denote the equivalences induced by composition with β_{G}^{c} and β^{∗}. Thus

c_{β}(L_{p}^{c}(BG_{p}^{∧}) ⟶^{ρ} L_{p}^{c}(BG_{p}^{∧})) = β^{∗} ∘ ρ ∘ β_{G}^{c} and c_{β}^{∗}(L_{p}^{c}(G) ⟶^{ψ} L_{p}^{c}(G)) = β_{G}^{c} ∘ ψ ∘ β^{∗},

and similarly for morphisms. Note that c_{β}^{∗} preserves the monoidal structures of these categories (since β^{∗} ∘ β_{G}^{c} = Id by assumption), while c_{β} does not in general preserve them. Let L = L_{G} to be the composite

L = L_{G}: Aut(BG_{p}^{∧}) ⟶^{L_{BG^{∧}_{p}}} Aut_{typ}(L_{p}^{c}(BG_{p}^{∧})) ⟶_{≃}^{c_{β}} Aut_{typ}(L_{p}^{c}(G)).

It remains to show that the maps induced by R and L on nerves are mutual inverses up to natural isomorphism of functors, thus proving Theorems B and C. We first show that L ∘ R ≅ Id.

For each p-centric subgroup P ≤ G, let t_{P} : B(P) → L_{p}^{c}(G) denote the functor which sends the unique object o_{P} to P, and sends each morphism x̃ in B(P) to x̂ ∈ Aut_{L_{p}^{c}(G)}(P). For each pair of p-centric subgroups P, Q ≤ G and each x ∈ N_{G}(P, Q), let

H(x): t_{P} ⟶ t_{Q} ∘ B(c_{x})

be the natural morphism of functors B(P) → L_{p}^{c}(G) which sends the object o_{P} to

the morphism P x̂→ Q. This induces a homotopy

|H(x)|: BP × I = |B(P) × [1]| ⟶ |L_{p}^{c}(G)|;

i.e., a path in the mapping space Map(BP, |L_{p}^{c}(G)|). Define

Γ_{G}: L_{p}^{c}(G) ⟶ L_{p}^{c}(|L_{p}^{c}(G)|)

to be the functor which sends an object P to the pair (P, |t_{P} |), and which sends a morphism P \underrightarrow{x̂} Q in L_{p}^{c}(G) (for x ∈ N_{G}(P, Q)) to Γ_{G}(x̂) = (c_{x}, |H(x)|).

The next two lemmas explore some of the features of Γ_{G}.

Lemma 3.5. For any finite group G, the following square

\begin{matrix}L_{p}^{c}(G) & \overline{^{β_{G}^{c}} →} & L_{p}^{c}(BG_{p}^{∧}) \\ L_{p}^{c}(|L_{p}^{c}(G)|) & ↓^{Γ_{G}} & \overline{^{L_{p}^{c}(κ)} →} & L_{p}^{c}(|L_{p}^{c}(G)|_{p}^{∧}) & ^{↑^{L_{p}^{c}(BG_{p}^{∧})}} \end{matrix}

commutes up to natural isomorphism of functors.

Proof. Consider the following commutative diagram

\xymatrix{ & L_{p}^{c}(BG) \ar[r]^{L_{p}^{c}(κ)} & L_{p}^{c}(BG_{p}^{∧}) \\ L_{p}^{c}(G) \ar[r]^{Γ_{G}} \ar[ur]^{β_{G}} \ar[d]_{π} & L_{p}^{c}(|L_{p}^{c}(G)|) \ar[r]^{L_{p}^{c}(κ)} \ar[u]_{L(α_{G})} \ar[d]^{L(|π|)} & L_{p}^{c}(|L_{p}^{c}(G)|_{p}^{∧}) \ar[u]^{≃}_{L((α_{G})_{p}^{∧})} \ar[d]_{≃}^{L(|π|_{p}^{∧})} \\ L_{p}^{c}(G) \ar[r]^{_{Γ_{G}}} & L_{p}^{c}(|L_{p}^{c}(G)|) \ar[r]^{L_{p}^{c}(κ)} & L_{p}^{c}(|L_{p}^{c}(G)|_{p}^{∧})}

of functors between categories. Here Γ_{G} is defined in a way analogous to Γ_{G}. Since

L((α_{G})_{p}^{∧}) ≅ L(α_{G}) ∘ L(|π|_{p}^{∧})

by Proposition 1.1, this shows that the two composites

L_{p}^{c}(G) ⟶^{π} L_{p}^{c}(G) \overrightarrow{⟶}_{β_{G}^{c}}^{L_{p}^{c}(α_{G})∘L_{p}^{c}(κ)∘Γ_{G}} L_{p}^{c}(BG_{p}^{∧})

are naturally isomorphic. And since π induces a bijection on objects and a surjection on morphisms, any natural isomorphism of functors after composition with π (when regarded as a map from Ob(L_{p}^{c}(G)) = Ob(L_{p}^{c}(G)) to Mor(L_{p}^{c}(BG_{p}^{∧}))) is also a natural isomorphism before composition.

The second property we need of the functors Γ_{G} is their naturality with respect to isotypical equivalences.

Lemma 3.6. For any isotypical equivalence ψ: L_{p}^{c}(G) → L_{p}^{c}(G), the following square

\begin{matrix}L_{p}^{c}(G) & \overline{^{Γ_{G}}→} & L_{p}^{c}(|L_{p}^{c}(G)|) \\ L_{p}^{c}(G) & ↓^{ψ} & \overline{^{Γ_{G}}→} & L_{p}^{c}(|L_{p}^{c}(G)|); & ↓^{L_{p}^{c}(|ψ|)} \end{matrix}  (1)

commutes up to a natural isomorphism of functors

W (ψ): L_{p}^{c}(|ψ|) ∘ Γ_{G} ⟶^{≅} Γ_{G} ∘ ψ,

which is itself natural in the sense that the following square commutes for any natural isomorphism Ψ: ψ ^{≅}→ ψ′:

L_{p}^{c}(|ψ|) ∘ Γ_{G} \overline{→^{W (ψ)}} Γ_{G} ∘ ψ

L_{p}^{c}(|ψ′|) ∘ Γ_{G} ↓^{L_{p}^{c}(|Ψ|)∘Γ_{G}}  \overline{→^{W (ψ′)}} Γ_{G} ∘ ψ′. ↓^{Γ_{G}∘Ψ}

(2)

Proof. We first claim that

W (ψ) : L_{p}^{c}(|ψ|) ∘ Γ_{G} ⟶ Γ_{G} ∘ ψ,

defined by sending an object P in L_{p}^{c}(G) to the morphism

(ψ_{P} , C_{P}) ∈ Mor_{L_{p}^{c}(|L_{p}^{c}(G)|)}((P, |ψ| ∘ |t_{P} |), (P′, |t_{P′}|)),

is a natural isomorphism of functors. Here, P′ = ψ(P) for short, and C_{P} denotes the constant homotopy. Note first that the objects are correct: L_{p}^{c}(|ψ|)(Γ_{G}(P)) = (P, |ψ| ∘ |t_{P} |) and Γ_{G}(ψ(P)) = (P′, |t_{P′}|) by definition. Also, (ψ_{P} , C_{P}) is a morphism between these objects, since

|ψ| ∘ |t_{P} | = |ψ ∘ t_{P} | = |t_{P′} ∘ B(ψ_{P})| = |t_{P′}| ∘ Bψ_{P}

by definition of ψ_{P}. To show that W (ψ) is natural, we must check, for each morphism x̂: P → Q in L_{p}^{c}(G), with ŷ = ψ(x̂) and Q′ = ψ(Q), that the following square commutes:

\begin{matrix}(P, |ψ| ∘ |t_{P} |) & \overline{→^{(ψ_{P} ,C_{P})}} & (P′, |t_{P′}|) \\ (Q, |ψ| ∘ |t_{Q}|) & ↓^{(c_{x},|ψ|∘|H(x)|)} & \overline{→^{(ψ_{Q},C_{Q})}} & (Q′, |t_{Q′}|) & ↓^{(c_{y},|H(y)|)} \end{matrix}

in L_{p}^{c}(|L_{p}^{c}(G)|). By Lemma 3.4, (P ↦ ψ_{P}) is a natural isomorphism of functors

λ_{G} ^{≅}→ λ_{G} ∘ψ, and thus c_{y} ∘ψ_{P} = ψ_{Q} ∘c_{x}. So it remains to show that |H(y)| ∘(Bψ_{P} × I) = |ψ| ∘ |H(x)|; and this follows since both are induced by the natural isomorphism (o_{P} ↦ ŷ) of functors B(P) ⟶ L_{p}^{c}(G).

It remains to check that square (2) commutes for all Ψ : ψ ^{≅}→ ψ′. This means showing, for each p-centric P ≤ G, and z such that ẑ = Ψ_{P} ∈ Iso_{L_{p}^{c}(G)}(ψ(P), ψ′(P)),

that the following square commutes in L_{p}^{c}(|L_{p}^{c}(G)|):

\begin{matrix}(P, |ψ| ∘ |t_{P} |) & \overline{→^{(ψ_{P} ,C_{P})}} & (ψ(P), |t_{ψ(P)}|) \\ (P, |ψ′| ∘ |t_{P} |) & ↓^{(Id_{P} ,|Ψ|∘|t_{P} |)} & \overline{→^{(ψ′_{P} ,C_{P})}} & (ψ′(P), |t_{ψ′(P)}|). & ↓^{Γ_{G}(Ψ_{P}) =(c_{z}, |H(z)|)} \end{matrix}

The square of group homomorphisms commutes (c_{z} ∘ ψ_{P} = ψ′_{P}) since for each g ∈ P, the square

\begin{matrix}ψ(P) & →^{{ \atop ψ_{P} (g)}=ψ_{P,P} (ĝ)} & ψ(P) \\ ψ′(P) & ↓^{Ψ_{P} =ẑ ≅} & \overline{→^{{ \atop ψ′_{P} (g)}=ψ′_{P,P} (ĝ)}} & ψ′(P) & ↓^{≅ Ψ_{P} =ẑ} \end{matrix}

commutes by naturality of Ψ. Since the C_{P} are constant homotopies, it remains to check that |Ψ| ∘ |t_{P} | and |H(Ψ_{P})| ∘ (Bψ_{P} × I) are homotopic as paths in the space Map(BP, |L_{p}^{c}(G)|) from |ψ| ∘|t_{P} | = |t_{ψ(P)}| ∘Bψ_{P} to |ψ′| ∘|t_{P} | = |t_{ψ′(P)}| ∘Bψ′_{P}. And this holds since both are induced by the natural isomorphism of functors B(G) → L_{p}^{c}(G) which sends the object o_{P} to the morphism Ψ_{P}.

We are now ready to show:

Proposition 3.7. For any finite group G, the composite

Aut_{typ}(L_{p}^{c}(G)) ⟶^{R} Aut(BG_{p}^{∧}) ⟶^{L} Aut_{typ}(L_{p}^{c}(G))

is naturally isomorphic to the identity.

Proof. Fix an isotypical equivalence ψ : L_{p}^{c}(G) → L_{p}^{c}(G), and consider the following diagram:

\xymatrix{ & L_{p}^{c}(G) \ar[rrr]^{ψ} \ar[dl]_{_{Γ_{G}}} \ar@{-->}[dr]^{_{β_{c}_{G}}}_{≃} &  &  & L_{p}^{c}(G) \ar[dr]^{_{Γ_{G}}} \\ L_{p}^{c}(|L_{p}^{c}(G)|) \ar[dr]_{L_{p}^{c}(κ)} &  & L_{p}^{c}(BG_{p}^{∧}) \ar[r]^{L_{p}^{c}(R(ψ))} \ar@{-->}[dl]_{≃}^{L_{p}^{c}(α^{∗})} & L_{p}^{c}(BG_{p}^{∧}) \ar[ur]_{≃}^{β^{∗}} &  & L_{p}^{c}(|L_{p}^{c}(G)|) \ar[dl]^{L_{p}^{c}(κ)} \\  & L_{p}^{c}(|L_{p}^{c}(G)|_{p}^{∧}) \ar[rrr]^{L_{p}^{c}(|ψ|_{p}^{∧})} &  &  & L_{p}^{c}(|L_{p}^{c}(G)|_{p}^{∧}). \ar@{-->}[ul]_{≃}^{L_{p}^{c}(α_{G})}}

Here, α^{∗} and β^{∗} are the inverses (up to homotopy or natural isomorphism) of α_{G} and β_{G}^{c} used to define R and L. In particular, L ∘ R(ψ) = β^{∗} ∘ L_{p}^{c}(R(ψ)) ∘ β_{G}^{c}, and proving the proposition means showing that the upper trapezoid commutes up to natural isomorphism of functors. And this follows since the two squares in the diagram commute up to natural isomorphism by Lemma 3.5, the large hexagon by Lemma 3.6, and the lower trapezoid (commutes precisely) by definition of R.

To make this more precise, fix natural isomorphisms of functors

U: L_{p}^{c}(α^{∗}) ∘ β_{G}^{c} ⟶ L_{p}^{c}(κ) ∘ Γ_{G}

and

V : β^{∗} ∘ L_{p}^{c}(α_{G}) ∘ L_{p}^{c}(κ) ∘ Γ_{G} ⟶ Id

(chosen independently of ψ), and let

W _{p}^{∧}(ψ) : L_{p}^{c}(|ψ|_{p}^{∧}) ∘ L_{p}^{c}(κ) ∘ Γ_{G} ⟶ L_{p}^{c}(κ) ∘ Γ_{G} ∘ ψ

be the natural isomorphism of Lemma 3.6 after composing with completion at p. Define A(ψ) : L ∘ R(ψ) → ψ to be the composite of natural isomorphisms

L ∘ R(ψ) =^{def} β^{∗} ∘ L_{p}^{c}(Rψ) ∘ β_{G}^{c} = β^{∗} ∘ L_{p}^{c}(α_{G}) ∘ L_{p}^{c}(|ψ|_{p}^{∧}) ∘ L_{p}^{c}(α^{∗}) ∘ β_{G}^{c}

⟶^{(−)∘U} β^{∗} ∘ L_{p}^{c}(α_{G}) ∘ L_{p}^{c}(|ψ|_{p}^{∧}) ∘ L_{p}^{c}(κ) ∘ Γ_{G}

⟶^{(−)∘W _{p}^{∧}(ψ)} β^{∗} ∘ L_{p}^{c}(α_{G}) ∘ L_{p}^{c}(κ) ∘ Γ_{G} ∘ ψ ⟶^{V ∘ψ} ψ.

To see that A : L ∘ R → Id is an isomorphism of functors, it remains only to check its naturality with respect to morphisms in Aut_{typ}(L_{p}^{c}(G)). And since U and V are independent of ψ, the naturality of A follows from the naturality of W _{p}^{∧}(ψ) with respect to isomorphisms Ψ : ψ → ψ′, as shown in Lemma 3.6.

It remains to show that R ∘L is naturally equivalent to the identity functor. Notice that the map L_{G} is defined as the composite c_{β} ∘ L_{BG_{p}^{∧}}. Thus, to prove this claim, it suffices to show that the functor |L_{BG_{p}^{∧}}| induces a monomorphism on homotopy groups. One way to do this is via a homology decomposition of BG, using standard techniques to construct maps on a homotopy colimit. However, we give a different argument in the next section: one which uses the techniques already developed in this paper.

4. Proof of Theorems B and C

We now finish the proof that Aut(BG_{p}^{∧}) has the homotopy type of |Aut_{typ}(L_{p}^{c}(G))|,

by showing that Aut(BG_{p}^{∧}) ^{L}→ Aut_{typ}(L_{p}^{c}(G)) is homotopy split injective. This

could be done more directly if there were natural “evaluation” maps L_{p}^{c}(X) → X,

induced by evaluating each BP → X at the basepoint of BP. Since there are no such natural maps, we instead define a simplicial space Mc_{•}(X), whose topological realization |Mc_{•}(X)| has the mod-p homotopy type of |L_{p}^{c}(X)| when X = BG or BG_{p}^{∧}, and which is equipped with a natural evaluation map to X. Note that we use | − | to denote both the topological realization of a simplicial space and the topological realization of the nerve of a category.

To help motivate this construction, note the following analogy between L_{p}(X) and the fundamental groupoid π(X), which can be thought of as the full subcategory of L_{p}(X) whose objects are the trivial subgroups (pt → X). The realization |π(X)| does not have a (natural) evaluation map to X, but there is an obvious map |S_{•}(X)| ^{ev}→ X defined on the realization of the singular simplicial set. There is also a natural map |S_{•}(X)| → |π(X)| which sends a singular simplex (Δ^{n} → X) to the sequence of paths obtained by restriction to the edges [v_{i}, v_{i+1}], and this map is a homotopy equivalence if X is aspherical. By analogy, Mc_{•}(X) should be a simplicial space whose n-simplices are maps on Δ^{n} which send each point to some mapping space Map(BP, X) with appropriate continuity conditions. In practice, it is simpler to define Mc_{•}(X) using certain spaces Δ(P) defined as follows.

For any sequence

P = (P_{0} ^{φ_{1}}→ P_{1} ^{φ_{2}}→ ⋯ ^{φ_{n}}→ P_{n})

of finite p-groups and monomorphisms, define the space Δ(P) inductively as follows. Set Δ(P_{0}) = BP_{0}, let Δ(P_{0} ⟶^{φ_{1}} P_{1}) be the mapping cylinder of the map BP_{0} ^{Bφ_{1}}→ BP_{1}, and in general let Δ(P) be the mapping cylinder of the map from Δ(P_{0} → ⋯ → P_{n−1}) to BP_{n} induced by the Bφ_{i}. Upon identifying the n-simplex Δ^{n} with the mapping cylinder of (Δ^{n−1} → pt), the inclusions of basepoints into the BP_{i} and the projections to points induce maps

Δ^{n} ⟶^{ι_{P}} Δ(P) ⟶^{ω_{P}} Δ^{n}.

Let {v_{0}, …, v_{n}} denote the vertices of Δ^{n}, and write BP_{i}×v_{i} (≅ BP_{i}) to denote ω_{P}^{−1}(v_{i}).

Equivalently, for P as above, we can define

Δ(P) = (∐_{i=0}^{n}(BP_{i} × Δ^{n−i}))/∼,

where the equivalence relation is defined by

(x, d_{0}(t)) ∼ (Bφ_{i}(x), t) for all x ∈ BP_{i−1}, t ∈ Δ^{n−i}.

Here, d_{0} : Δ^{n−i} → Δ^{n−i+1} is the face map which sends the vertex v_{j} (for 0 ≤ j ≤ i−1) to v_{j+1}. Under this identification, Δ^{n} ^{ι_{P}}→ Δ(P) is induced by the inclusion of the basepoint in BP_{0}, and Δ(P) ^{ω_{P}}→ Δ^{n} is induced by projection to the second factor followed by the face maps Δ^{n−i} → Δ^{n} which send v_{j} to v_{j+i}.

Alternatively, Δ(P) can be regarded as the homotopy colimit of the sequence

BP_{0} → BP_{1} → ⋯ → BP_{n},

regarded as a functor on the poset category [n] = {0 < 1 < ⋯ < n}. Or it can be thought of as the realization of the simplicial space Δ_{•}(P), where

Δ_{k}(P) = ∐_{0≤i_{0}≤···≤i_{k}≤n} BP_{i_{0}},

with the obvious face and degeneracy maps.

Definition 4.1. For any space X, let Mc_{•}(X) be the simplicial space defined as follows. The vertices in Mc_{•}(X) are the p-centric subgroups (P, α) in X. An n-simplex η ∈ M_{n}^{c}(X) is a map

η : Δ(P) ⟶ X,

for some sequence P = (P_{0} → ⋯ → P_{n}) of finite p-groups and monomorphisms, such that η|_{(BP_{i}×v_{i})} is a p-centric subgroup of X for each i. This n-simplex spans the vertices (P_{i}, η|_{(BP_{i}×v_{i})}) for i = 0, …, n. The set of vertices M_{0}^{c}(X) is given the discrete topology. The set of n-simplices spanning a given set of vertices (P_{i}, α_{i}) and based on a given sequence of monomorphisms P has the compact-open topology, and is open in the space of all n-simplices.

The face and degeneracy maps in Mc_{•}(X) are described as follows. For any P = (P_{0} → ⋯ → P_{n}) and any morphism \underline{m} ^{γ}→ \underline{n} in the simplicial category, let γ^{∗}P denote the sequence P_{γ(0)} → ⋯ → P_{γ(m)}, and let

Δ(γ, P): Δ(γ^{∗}P) ⟶ Δ(P)

denote the map defined by sending (BP_{γ(i)} × v_{i}) to (BP_{γ(i)} × v_{γ(i)}) (for i = 0, …, m) and extending linearly. Then M_{n}^{c}(X) ^{γ^{∗}}→ M_{m}^{c}(X) sends an n-simplex Δ(P) ^{η}→ X to the m-simplex η ∘ Δ(γ, P).

In fact, wherever Mc_{•}(X) is used throughout this section, it will be necessary to replace it by its “levelwise” singular simplicial set S_{•}Mc_{•}(X). In other words, this is the bisimplicial set where each space M_{n}^{c}(X) is replaced by its singular simplicial set S_{•}M_{n}^{c}(X). In general, this replacement goes through without problems; we make a few comments at key points in the arguments where this can make a difference in statements or definitions.

For each n-simplex Δ(P) = Δ(P_{0} → ⋯ → P_{n}) ^{η}→ X in Mc_{•}(X), the restriction of η to each mapping cylinder ω_{P}^{−1}(⟨v_{i}, v_{i+1}⟩) ≅ Δ(P_{i} → P_{i+1}) determines a morphism in L_{p}^{c}(X). The n-simplex η thus determines an n-simplex in the nerve of L_{p}^{c}(X), regarded as a simplicial space. This induces a map of simplicial spaces

τ_{X}: Mc_{•}(X) ⟶ N_{•}(L_{p}^{c}(X)),

where N_{•}(−) denotes the nerve of the category (as opposed to its topological realization).

In the following lemma, the simplicial space Mc_{•}(−) must in fact be replaced by S_{•}Mc_{•}(−) to get a “genuine” (not just weak) homotopy equivalence. This will be important in the proof of Proposition 4.4 (see the explanation after diagram (1) in that proof).

Lemma 4.2. If X = BG or BG_{p}^{∧} for a finite group G, then |Mc_{•}(X)| ^{|τ_{X}|} → |L_{p}^{c}(X)| is a homotopy equivalence.

Proof. When X = BG or BG_{p}^{∧}, then for any p-centric subgroup BP ^{α} → X, Map(BP, X)_{α} is aspherical by Proposition 2.1. We claim that |τ_{X}| is a homotopy equivalence for any X with this property.

Fix a set of vertices (P_{i}, α_{i}), and a sequence of monomorphisms P = (P_{0} ⟶^{φ_{1}} ⋯ ⟶^{φ_{n}} P_{n}). Let Δ_{0}(P) be the union of the mapping cylinders of the Bφ_{i}, regarded as a subspace of Δ(P) (i.e., Δ_{0}(P) = ⋃_{i=1}^{n}(ω_{P}^{−1}[v_{i−1}, v_{i}])). Let Y denote the space of maps Δ(P) ^{η}→ X such that η|_{BP_{i}×v_{i}} = α_{i} for each i, and let Y_{0} denote the space of maps Δ_{0}(P) → X with the same property. Since Δ_{0}(P) is a deformation retract of Δ(P), the restriction map Y → Y_{0} is a homotopy equivalence. Also, π_{0}(Y_{0}) is the set of sequences of morphisms in L_{p}^{c}(X) of the form

(P_{0}, α_{0}) ⟶^{(φ_{1},η_{1})} (P_{1}, α_{1}) ⟶^{(φ_{2},η_{2})} ⋯ ⟶^{(φ_{n},η_{n})} (P_{n}, α_{n}).

In other words, the set of n-simplices in the nerve of L_{p}^{c}(X) is the set of connected components of the space of n-simplices in Mc_{•}(X) (this holds for any X), and it remains only to show that each component in Y_{0} is contractible. But Y_{0} is the product over all i of the spaces of paths in Map(BP_{i}, X) from α_{i} to α_{i+1} ∘ Bφ_{i+1}; and the connected components of these spaces are contractible since Map(BP_{i}, X)_{α_{i}} is aspherical by assumption.

This shows that τ_{X}, as a map of simplicial spaces, is a homotopy equivalence in each degree. So |τ_{X}| is a homotopy equivalence by [GJ, Proposition IV.1.7].

For any X, define maps

ev_{X,n}: M_{n}^{c}(X) × Δ^{n} ⟶ X,

for each n ≥ 0, by setting

ev_{X,n}(η, t) = η(ι_{P}(t))

for each Δ(P) ^{η}→ X in M_{n}^{c}(X) and each t ∈ Δ^{n}. In other words, this is the map defined by restriction to basepoints in the BP_{i} (recall that Δ^{n} ^{ι_{P}}→ Δ(P) is induced by inclusions of basepoints). The ev_{X,n} are continuous and commute with all face and degeneracy maps, and hence combine to define an evaluation map

ev_{X}: |Mc_{•}(X)| ⟶ X.

Lemma 4.3. If X = BG or BG_{p}^{∧} for some finite group G, then |Mc_{•}(X)| ^{ev_{X}}→ X is an 𝔽_{p}-homology equivalence.

Proof. Assume first that X = BG, and consider the following diagram:

\xymatrix{|L_{p}^{c}(G)| \ar[r]^{β_{G}^{c}}_{≃} \ar[dd]_{α_{G}}^{≃} & |L_{p}^{c}(BG)| \ar[dd]_{ϵ_{1}} & |Mc_{•}(BG)| \ar[l]_{|τ_{BG}|}^{≃} \ar[dd]_{ϵ_{2}} \ar@{-->}[dr]^{_{ev_{BG}}} \\  &  &  & BG. \\ |B(G)| \ar[r]^{γ_{G}}_{≃} & |π(BG)| & |Stop_{•}(BG)| \ar[l]_{|τ′|}^{≃} \ar@{-->}[ur]^{ϵ_{3}}_{≃}}  (1)

Here, Stop_{•}(BG) is the singular simplicial set for BG, but topologized so that the set of vertices is discrete, but the set of n-simplices Δ^{n} → BG with any given set of vertices is given the compact-open topology. The maps ϵ_{1} and ϵ_{2} are both defined by restricting maps BP → BG to the basepoint (in the case of ϵ_{2} via composition with ι_{P}). And ϵ_{3} is the evaluation map for singular simplices in BG. Also, τ′ is defined by sending each vertex in Stop_{•}(BG) (i.e., each point of BG) to itself considered as an object in π(BG), and by sending each singular simplex σ : Δ^{n} → BG to its homotopy class regarded as a sequence of paths in BG (i.e., an n-simplex in the nerve of π(BG)). And γ_{G} is induced by regarding B(G) as a subcategory of π(BG) — the full subcategory with object the basepoint ∗ of BG. The triangle in (1) commutes by definition of ev_{BG}, and both squares are induced by commutative squares of categories and functors or of simplicial spaces.

Now, |τ′| is a homotopy equivalence since τ′ is a map of simplicial spaces which is a homotopy equivalence at each level (the components of the space of n-simplices in Stop_{•}(BG) are contractible since BG is aspherical). The evaluation map ϵ_{3} is also a homotopy equivalence, since S_{n}^{top}(BG) is homotopy equivalent to BG for each n. Also, |τ_{BG}| is a homotopy equivalence by Lemma 4.2, β_{G}^{c} and α_{G} are homotopy equivalences by Propositions 2.6 and 1.1, and γ_{G} is an equivalence since it is induced by an equivalence of categories. The commutativity of the diagram now proves that ev_{BG} is a homotopy equivalence.

To see that ev_{BG_{p}^{∧}} is an 𝔽_{p}-homology equivalence, consider the following diagram, where the vertical maps are all induced by p-completion:

\begin{aligned}|L_{p}^{c}(BG)| & ← ^{|τ_{BG}|} & |Mc_{•}(BG)| & \overline{^{ev_{BG}} →} & BG \\ |L_{p}^{c}(BG_{p}^{∧})| & ↓^{L_{p}^{c}(κ)} & ←^{|τ_{BG^{∧}_{p}}|} & |Mc_{•}(BG_{p}^{∧})| & ↓^{M_{•}^{c}(κ)} & \overline{^{ev_{BG^{∧}_{p}}} →} & BG_{p}^{∧}. & ↓^{κ} \end{aligned}

The squares commute by naturality of τ and ev. Also, |L_{p}^{c}(κ)| is an 𝔽_{p}-homology equivalence by Corollary 2.8, and κ is clearly an equivalence. We have just shown that ev_{BG} is a homotopy equivalence, and |τ_{BG}| and |τ_{BG_{p}^{∧}}| are homotopy equivalences by Lemma 4.2. So ev_{BG_{p}^{∧}} is also an 𝔽_{p}-homology equivalence.

All of the definitions and results so far in this section were developed to prove the following result:

Proposition 4.4. For any finite group G, the map

|L_{BG_{p}^{∧}}| : |Aut(BG_{p}^{∧})| ⟶ |Aut_{typ}(L_{p}^{c}(BG_{p}^{∧}))|

induces a split monomorphism on π_{i} for all i ≥ 0.

Proof. For any pair of spaces X and Y, let Eqv(X, Y) denote the groupoid whose objects are the mod-p-equivalences X → Y, and whose morphisms are homotopy classes of homotopies. Clearly, composition with homotopy equivalences induces homotopy equivalences between the nerves of such categories.

To shorten the notation, we write Mc_{•} = Mc_{•}(BG_{p}^{∧}) and L_{p}^{c} = L_{p}^{c}(BG_{p}^{∧}). We will construct a functor ξ : Aut(BG_{p}^{∧}) → Aut(Mc_{•}) which makes the following diagram commute (precisely, not just up to natural isomorphism):

\xymatrix{ & Aut(BG_{p}^{∧}) \ar[r]^{L_{BG^{∧}_{p}}} \ar[dd]^{ξ} \ar@{-->}[dl]_{−∘ev}^{≃} & Aut(L_{p}^{c}) \ar@{-->}[dr]^{|−|} \\ Eqv(|Mc_{•}|_{p}^{∧}, BG_{p}^{∧}) &  &  & Aut(|L_{p}^{c}|_{p}^{∧}) \ar@{-->}[dl]^{−∘|τ|}_{≃} \\  & Aut(|Mc_{•}|_{p}^{∧}) \ar[r]^{|τ|∘−}_{≃} \ar@{-->}[ul]^{ev∘−}_{≃} & Eqv(|Mc_{•}|_{p}^{∧}, |L_{p}^{c}|_{p}^{∧})}  (1)

The lemma then follows immediately, since the maps labeled “≃” are all induced by composition with homotopy equivalences (and hence themselves induce homotopy equivalences on nerves).

In fact, when carrying out this argument, it is essential to replace |Mc_{•}(BG_{p}^{∧})| by |S_{•}Mc_{•}(BG_{p}^{∧})| in the above diagram. This is necessary to guarantee that the maps

BG_{p}^{∧} ⟵^{|τ|} |S_{•}Mc_{•}(BG_{p}^{∧})|_{p}^{∧} ⟶^{|ev|} |L_{p}^{c}(BG_{p}^{∧})|_{p}^{∧}

are homotopy equivalences (not just weakly), and hence that composition with |τ| or |ev| is a homotopy equivalence.

We define ξ as follows. On objects, ξ sends a homotopy equivalence BG_{p}^{∧} ^{f}→ BG_{p}^{∧}

to the induced functor |Mc_{•}(BG_{p}^{∧})| \begin{matrix}ξ(f) \\ =|M_{•}^{c}(f)| \end{matrix} → |Mc_{•}(BG_{p}^{∧})|, defined via composition with f. So it remains to define ξ(F) : |Mc_{•}| × I → |Mc_{•}| for each homotopy F : BG_{p}^{∧} ×

I → BG_{p}^{∧}. To do this, we regard I as the realization of the simplicial set [1], where [1]_{n} is the set of sequences σ = (σ(0) ≤ ⋯ ≤ σ(n)) taking values in {0, 1}. For any such σ, we let Δ^{n} ^{|σ|}→ I denote the affine map which sends v_{i} to σ(i). Let

ξ_{•}(F): Mc_{•} × [1] → Mc_{•}

be the map of simplicial spaces which sends the n-simplex (Δ(P) ^{η}→ BG_{p}^{∧}, σ) in Mc_{•} × [1] to the n-simplex

F (η, σ): Δ(P) ⟶^{(Id,|σ|∘ω_{P})} Δ(P) × I ⟶^{η×I} BG_{p}^{∧} × I ⟶^{F} BG_{p}^{∧},

where ω_{P} : Δ(P) → Δ^{n} is the map defined at the beginning of the section. The F (η, σ) are easily seen to be continuous and to commute with face and degeneracy maps, and hence define a homotopy ξ(F) = |ξ_{•}(F)|.

Note that an obvious modification in the above definition of ξ gives a functor which takes values in the automorphism category Aut(|S_{•}Mc_{•}(BG_{p}^{∧})|_{p}^{∧}).

The commutativity of (1) follows from the commutativity of the following squares, for any homotopy F : BG_{p}^{∧} × I → BG_{p}^{∧}:

\begin{aligned}BG_{p}^{∧} × I & ←^{ev×I} & |Mc_{•}(BG_{p}^{∧})| × I & →^{τ×I} & |L_{p}^{c}(BG_{p}^{∧})| × I \\ BG_{p}^{∧} & ↓^{F} & ← ^{ev} & |Mc_{•}(BG_{p}^{∧})| & ↓^{ξ_{•}(F)} & \overline{^{τ} →} & |L_{p}^{c}(BG_{p}^{∧})|. & ↓^{L_{BG^{∧}_{p}}(F)} \end{aligned}

And this is easily checked.

We are now ready to prove Theorems B and C. In particular, we want to show that Aut(BG_{p}^{∧}) and |Aut(L_{p}^{c}(G))| are homotopy equivalent, and moreover equivalent as monoids. There is no obvious way to construct a map between these two spaces which is both a homotopy equivalence and a morphism of monoids, so instead we connect them via a sequence of maps. Here S_{•}X, for any space X, denotes the (discrete) singular simplicial set of X.

Theorem 4.5. For any finite group G and any prime p, Aut(BG_{p}^{∧}) has the homotopy type of |Aut_{typ}(L_{p}^{c}(G))|. More precisely:

(a) R and L induce isomorphisms

Out_{typ}(L_{p}^{c}(G)) \overleftarrow{⟶}_{π_{0}R}^{π_{0}L} Out(BG_{p}^{∧})

between the groups of components.

(b) The following maps are morphisms of topological monoids and homotopy equivalences:

Aut(BG_{p}^{∧}) ⟵_{≃}^{ev} |S_{•} Aut(BG_{p}^{∧})| ⟶_{≃}^{σ} |Aut(BG_{p}^{∧})|

⟶_{≃}^{|L_{BG^{∧}_{p}}|} |Aut_{typ}(L_{p}^{c}(BG_{p}^{∧}))| ⟵_{≃}^{|c_{β}^{∗}|} |Aut_{typ}(L_{p}^{c}(G))|,

where σ is the map defined in Section 3. In particular,

BAut(BG_{p}^{∧}) ≃ B(|Aut_{typ}(L_{p}^{c}(G))|).

Proof. Consider the maps

|Aut_{typ}(L_{p}^{c}(G))| \overleftarrow{⟶}_{|R|}^{|L|} |Aut(BG_{p}^{∧})|

defined in Section 4. By Proposition 3.7, L ∘ R is naturally isomorphic to the identity on Aut_{typ}(L_{p}^{c}(G)); while |L| (equivalently |L_{BG_{p}^{∧}}|) is split injective on homotopy groups by Proposition 4.4. So R and L induce homotopy equivalences between these nerves; and in particular an isomorphism of groups between Out(BG_{p}^{∧}) = π_{0}(|Aut(BG_{p}^{∧})|) and Out_{typ}(L_{p}^{c}(G)) = π_{0}(|Aut_{typ}(L_{p}^{c}(G))|).

All four of the above maps in (b) are morphisms of monoids by construction, and ev and σ are homotopy equivalences by Lemma 3.1. Also, |c_{β}^{∗}| is a homotopy equivalence since it is induced by composition with equivalences between the two categories involved. Finally, |L_{BG_{p}^{∧}}| induces surjections on all homotopy groups by Proposition 3.7, and injections by Proposition 4.4.

The last statement now follows since a morphism of monoids which is a homotopy equivalence induces a homotopy equivalence between the classifying spaces, using [GJ, Proposition IV.1.7].

5. Higher limit obstructions

We now look more closely at ways of constructing isotypical equivalences between the categories L_{p}^{c}(G) and L_{p}^{c}(G′) when G and G′ are finite groups. We construct obstructions to the existence and uniqueness of such equivalences which “extend” given fusion preserving isomorphisms between Sylow p-subgroups. This is then used to prove Conjecture D whenever G has p-rank less than p^{2}.

Throughout this section and the next, for any functor ψ : C → C′ and any objects c, d in C, we let ψ_{c,d} denote the induced map from Mor_{C}(c, d) to Mor_{C′}(ψ(c), ψ(d)).

Let G and G′ be finite groups with Sylow p-subgroups S and S′ respectively, and assume S ≅ S′. Recall that an isomorphism S ^{f}→ S′ is called fusion preserving if given subgroups P, Q ≤ S and an isomorphism P ^{α}→ Q between them, α is induced by conjugation in G if and only if f(P) ^{f ∘α∘f^{−1}} → f(Q) is induced by conjugation in G′. The following lemma shows in particular that each equivalence ψ : L_{p}^{c}(G) → L_{p}^{c}(G′) determines a fusion preserving isomorphism f_{ψ} between Sylow p-subgroups of G and G′.

Lemma 5.1. Fix finite groups G and G′, and an isotypical equivalence

ψ : L_{p}^{c}(G) ⟶ L_{p}^{c}(G′)

of categories. Let S ≤ G and S′ = ψ(S) ≤ G′ be Sylow p-subgroups. Then

N_{G}(S)/C′_{G}(S) = Aut_{L_{p}^{c}(G)}(S) ⟶_{≅}^{ψ_{S,S}} Aut_{L_{p}^{c}(G′)}(S′) = N_{G′}(S′)/C′_{G′}(S′)

restricts to a fusion preserving isomorphism f_{ψ} : S ^{≅}→ S′. Furthermore:

(i) ψ is naturally isomorphic to an equivalence ψ′ which agrees with ψ on S and on Aut_{L_{p}^{c}(G)}(S) (hence f_{ψ′} = f_{ψ}), and which sends inclusions of subgroups of S to

inclusions of subgroups of S′. In other words, for any P ≤ S, ψ′(P ˆ1→ S) =

(ψ′(P) ˆ1→ S′).

(ii) Assume that ψ sends inclusions of subgroups of S to inclusions of subgroups of S′. Then for any p-centric P ≤ S, ψ(P) = f_{ψ}(P), and ψ_{P,P} sends ĝ ∈ Aut_{L_{p}^{c}(G)}(P)

to { \atop f_{ψ}(g)} ∈ Aut_{L_{p}^{c}(G′)}(ψ(P)) for all g ∈ P.

Proof. Since S ◁ N_{G}(S)/C′_{G}(S) and S′ ◁ N_{G′}(S′)/C′_{G′}(S′) are (unique) Sylow psubgroups, ψ_{S,S} restricts to an isomorphism f_{ψ} : S ^{≅}→ S′. We prove that f_{ψ} is fusion preserving as point (iii) below.

(i) For each p-centric P ≤ G, define a subgroup ψ′(P) ≤ S′ and an isomorphism α_{P} : ψ(P) ^{≅}→ ψ′(P) in L_{p}^{c}(G′) as follows. If P ≨ S, let y ∈ N_{G′}(ψ(P), S′) be such

that ψ_{P,S}(P \underrightarrow{ˆ1} S) = ŷ, and set ψ′(P) = y(ψ(P))y^{−1} (the “image” of ψ_{P,S}(ˆ1)) and α_{P} = ŷ. Otherwise, set ψ′(P) = ψ(P) and α_{P} = Id.

Now let ψ′ be the functor which sends P to ψ′(P), and where

ψ′_{P,Q}(P x̂→ Q) = (ψ′(P) ^{α(Q)∘ψ(x̂)∘α(P)^{−1}} → ψ′(Q)).

Then ψ′ is well defined as a functor, α is a natural isomorphism from ψ to ψ′, and ψ′ sends inclusions into S to inclusions into S′. Notice that ψ′ is automatically an isotypical equivalence.

(ii) Now assume ψ sends inclusions to inclusions. Fix a p-centric subgroup P ≤ S, and set P′ = ψ(P) ≤ S′ for short. For any element g ∈ P, the following square commutes in L_{p}^{c}(G′)

\begin{matrix}P′ & \overline{→^{ψ(ĝ)=ĥ}} & P′ \\ S′ & ˆ1 ; ↓ & \overline{→^{ψ(ĝ)={ \atop f_{ψ}(g)}}} & S′ & ˆ1 ; ↓ \end{matrix}

(for some h ∈ N_{G′}(P′)) since it is the image under ψ of a commutative square in

L_{p}^{c}(G). Thus h ≡ f_{ψ}(g) (mod C′_{G′}(P′)), so f_{ψ}(g) ∈ N_{G′}(P′), and ψ_{P,P} (ĝ) = { \atop f_{ψ}(g)}. In

particular, since ψ is isotypical, this shows that P′ = f_{ψ}(P).

(iii) It remains to show that f_{ψ} is fusion preserving. This is an easy consequence of Alperin’s fusion theorem in the form [Al, Theorem 5.1]; but since our result is much more elementary we give a direct proof.

For each P, let P θ_{P} ; ≅→ P′ be the restriction of ψ_{P,P} to P ≤ N_{G}(P)/C′_{G}(P). By

Lemma 3.4, this defines a natural isomorphism of functors θ: λ_{G} ^{≅}→ λ_{G} ∘ψ. Thus f_{ψ} may be identified with θ_{S} : S → S′. By (i), we can assume that ψ sends inclusions of subgroups in S to inclusions of subgroups in S′. It follows that θ_{P} : P → P′ is given by the restriction of θ_{S} to P. If P, Q ≤ S are p-centric in G and α = c_{x} : P → Q is a group isomorphism with x ∈ N_{G}(P, Q) then

f_{ψ}αf_{ψ}^{−1} = θ_{Q}αθ_{P}^{−1} = ψ(α) = c_{x′}

for some x′ ∈ N_{G′}(P′, Q′). Hence fusion is preserved among p-centric subgroups of S and S′.

Fix a subgroup P ≤ S which is not p-centric. We may assume inductively that f_{ψ} preserves fusion among all subgroups of strictly larger order. Let P be the set of subgroups of S which are G-conjugate to P. We may assume that N_{S}(P) is a Sylow p-subgroup of N_{G}(P) (replace P by some other subgroup in P if necessary).

Let P ≤ S be a Sylow p-subgroup of C_{G}(P)·P. Then

P ≨ P ≤ N_{G}(P)  (1)

since P is not p-centric, and we claim that

N_{G}(P) = (N_{G}(P) ∩ N_{G}(P))·C_{G}(P).  (2)

To see this, notice first that the right hand side is clearly a subgroup of N_{G}(P). The other inclusion follows because if x ∈ N_{G}(P), then xc ∈ N_{G}(P̄) for some c ∈ C_{G}(P). For any x ∈ N_{G}(P), c_{x} = c_{y} ∈ Aut(P) for some y ∈ N_{G}(P) ∩ N_{G}(P) by (2); f_{ψ} sends c_{y} ∈ Aut(P) to c_{y′} ∈ Aut(f_{ψ}(P)) for some y′ by (1) and the induction hypothesis; and hence sends c_{x} ∈ Aut(P) to c_{y′} ∈ Aut(f_{ψ}(P)). So fusion is preserved for automorphisms of P.

Now let Q be any other subgroup in P, and set R = N_{S}(Q) (≩ Q) for short. Since N_{S}(P) is a Sylow p-subgroup of N_{G}(P), there is x ∈ N_{G}(Q, P) such that xRx^{−1} ≤ N_{S}(P). By the induction hypothesis, there is x′ ∈ G′ such that f_{ψ} sends

R c_{x} ; ≅→ xRx^{−1} to f_{ψ}(R) c_{x′} ; ≅→ f_{ψ}(xRx^{−1}). But then x′ ∈ N_{G}(f_{ψ}(Q), f_{ψ}(P)), and so

f_{ψ}(Q) and f_{ψ}(P) are conjugate in G′.

This shows that f_{ψ} sends fusion among subgroups of S to fusion among subgroups of S′. The same argument applied to f_{ψ}^{−1} shows the converse, and thus that f_{ψ} is fusion preserving.

The isomorphism f_{ψ} : S → S′ of Lemma 5.1 will be referred to as the underlying fusion preserving isomorphism of the equivalence ψ : L_{p}^{c}(G) → L_{p}^{c}(G′).

Martino and Priddy [MP, Corollary 1.2] showed that if BG_{p}^{∧} ≃ BG′_{p}^{∧}, then there is a fusion preserving isomorphism between Sylow p-subgroups of G and of G′; this is a consequence of the description of [BP, BG_{p}^{∧}] (shown here as Proposition 2.1). By Theorem A, an equivalence BG_{p}^{∧} ≃ BG′_{p}^{∧} determines an equivalence of categories L_{p}^{c}(G) ≃ L_{p}^{c}(G′), and Lemma 5.1 gives an algebraic explanation of why this determines a fusion preserving isomorphism.

We will need the following property of fusion preserving isomorphisms.

Lemma 5.2. Let f : S ^{≅}→ S′ be any fusion preserving isomorphism between Sylow subgroups S ≤ G and S′ ≤ G′ of finite groups G and G′. Then a subgroup P ≤ S is p-centric in G if and only if f(P) is p-centric in G′.

Proof. Let P ≤ S be any subgroup. Then P is G-conjugate to some Q ≤ S such that C_{S}(Q) is a Sylow p-subgroup of C_{G}(Q). If P is not p-centric in G then the same holds for Q, and C_{S}(Q) ≰ Q. So f(Q) is not p-centric, and f(P) is G′-conjugate to f(Q) since f is fusion preserving. The converse follows from the same argument applied to f^{−1}.

We now want to study the problem of constructing an equivalence of categories L_{p}^{c}(G) → L_{p}^{c}(G′) which has a given isomorphism between Sylow p-subgroups as its underlying fusion preserving isomorphism. The obstructions to doing this lie in certain higher derived functors of inverse limits over orbit categories. The following is a key lemma in the construction.

Lemma 5.3. Let G and G′ be finite groups with Sylow p-subgroup S and S′ respectively. Let f : S → S′ be a fusion preserving isomorphism. Then for any P, Q ≤ S p-centric in G, there is a map

Φ_{P,Q}: N_{G}(P, Q) ⟶ N_{G′}(f(P), f(Q))

which satisfies the following three conditions for all x ∈ N_{G}(P, Q) (all x such that xP x^{−1} ≤ Q):

(a) the following square commutes (in the category of groups):

\begin{matrix}P & \overline{^{c_{x}} →} & Q \\ f(P) & ↓^{f} & \overline{^{c_{x′}} →} & f(Q), & ↓^{f} \end{matrix}

where x′ = Φ_{P,Q}(x).

(b) Φ_{P,Q}(gx) = f(g)Φ_{P,Q}(x) for all g ∈ Q.

(c) Φ_{P,Q} induces a bijection

Φ̂_{P,Q}: N_{G}(P, Q)/C′_{G}(P) ⟶^{≅} N_{G′}(f(P), f(Q))/C′_{G′}(f(P)).

Proof. To simplify notation, for any P ≤ S we write P′ = f(P) ≤ S′. Conditions (a) and (c) mean that there is a commutative diagram of the following form

N_{G}(P, Q) \overline{↠} N_{G}(P, Q)/C′_{G}(P) \overline{↠} N_{G}(P, Q)/C_{G}(P) ⊆ Hom(P, Q)

N_{G′}(P′, Q′) ↓^{Φ_{P,Q}}  ↠ N_{G′}(P′, Q′)/C′_{G′}(P′) ↓^{Φ̂_{P,Q} ≅}  ↠ N_{G′}(P′, Q′)/C_{G′}(P′) ↓^{≅}  ⊆ Hom(P′, Q′), ↓^{f_{∗} ≅}

where f_{∗} is the bijection between homomorphism sets induced by the isomorphisms P ≅ P′ and Q ≅ Q′. The restriction of f_{∗} to N_{G}(P, Q)/C_{G}(P) is defined and bijective by the assumption that f is fusion preserving.

Let X ⊆ N_{G}(P, Q) be a set of orbit representatives for the left action of Q and the right action of C′_{G}(P). For each x ∈ X, choose an arbitrary element x′ = Φ_{P,Q}(x) such that c_{x′} = f_{∗}(c_{x}). Extend this to all of N_{G}(P, Q) by setting Φ_{P,Q}(gxa) = f(g)Φ_{P,Q}(x) for all x ∈ X, g ∈ Q, and a ∈ C′_{G}(P). Then Φ_{P,Q} satisfies (a) and (b) above, and induces a map Φ̂_{P,Q} which makes the above diagram commute. Furthermore, Φ̂_{P,Q} is bijective, since the third column of the diagram is obtained by dividing out by the free action of Z(P) ≅ Z(P′) on the second column.

For any small category C and any contravariant functor C → Ab, let C^{∗}(C; F) denote the chain complex

C^{n}(C; F) = ∏_{c_{0}→···→c_{n}} F (c_{0}),

with differentials ϕ ∈ C^{n−1}(C; F) ^{d}→ C^{n}(C; F) defined by

dϕ(c_{0} ⟶^{f} c_{1} → ⋯ → c_{n}) = f^{∗}ϕ(c_{1} → ⋯ → c_{n}) + ∑_{k=1}^{n}(−1)^{k}ϕ(c_{0} → ⋯ ĉ_{k} ⋯ → c_{n}).

Then \underleftarrow{lim}_{C}^{∗}(F) ≅ H^{∗}(C^{∗}(C; F), d) (cf. [GZ, Appendix II, Proposition 3.3], applied with M = Ab^{op}).

Recall the notation used in Section 4 for objects and maps in orbit categories. For any finite G, O_{p}^{c}(G) is the category whose objects are the p-centric subgroups P ≤ G, and where Mor_{O_{p}^{c}(G)}(P, Q) = Q\N_{G}(P, Q) for any objects P, Q ≤ G. The morphism corresponding to x ∈ N_{G}(P, Q) is denoted x^{∘}, and corresponds to the G-map G/P → G/Q which sends gP to gx^{−1}Q in the usual definition of O_{p}^{c}(G).

For any element g in a finite group G, we let g_{p} ∈ G denote its “p-part”. This is the unique element of p-power order in the cyclic subgroup ⟨g⟩ such that |g^{−1}g_{p}| is prime to p. Equivalently, it is the unique element of p-power order in G for which there is an element g′ ∈ G of order prime to p such that [g_{p}, g′] = 1 and g = g_{p}·g′. In particular, for any p-centric subgroup P ≤ G, the projection of C_{G}(P) = Z(P)×C′_{G}(P) to Z(P) sends each g ∈ C_{G}(P) to g_{p} ∈ Z(P).

Proposition 5.4. Fix finite groups G and G′, and Sylow p-subgroups S ≤ G and S′ ≤ G′. Let f : S → S′ be any fusion preserving isomorphism. Let Z : O_{p}^{c}(G)^{op} → p-Gr denote the functor which sends P to its center Z(P). Then there is an element ω(f) ∈ \underleftarrow{lim}^{2}_{O_{p}^{c}(G)}(Z) with the following property: there is an isotypical equivalence of

categories

ψ : L_{p}^{c}(G) ⟶^{≅} L_{p}^{c}(G′)

with f_{ψ} = f, if and only if ω(f) = 0.

Proof. Again, for any P ≤ S we write P′ = f(P) ≤ S′. Let L_{S}^{c}(G) ⊆ L_{p}^{c}(G) and O_{S}^{c}(G) ⊆ O_{p}^{c}(G) denote the full subcategories whose objects are the p-centric subgroups of G contained in S, and similarly for L_{S′}^{c}(G′) ⊆ L_{p}^{c}(G′). All three of these inclusions are equivalences of categories, since every p-centric subgroup of G is conjugate to a subgroup of S. It follows that \underleftarrow{lim}^{∗}_{O_{S}^{c}(G)}(Z) ≅ \underleftarrow{lim}^{∗}_{O_{p}^{c}(G)}(Z). Also, any

equivalence ψ : L_{S}^{c}(G) → L_{S′}^{c}(G′) extends to an equivalence L_{p}^{c}(G) → L_{p}^{c}(G′). So we can restrict our attention to these subcategories.

We define ψ on objects by setting ψ(P) = P′ for all P ≤ S p-centric in G. By Lemma 5.2, this defines a bijection between objects in L_{S}^{c}(G) and L_{S′}^{c}(G′). The problem is to define ψ on morphisms. Thus, our goal is to construct the obstruction class ω(f), and show that ψ can be well defined on morphisms if and only if ω(f) = 0.

Given a choice of the Φ_{P,Q}, for all p-centric P, Q ≤ S, satisfying conditions (a), (b), and (c) of Lemma 5.3, we now define an element

ω_{Φ} ∈ C^{2}(O_{S}^{c}(G); Z) = ∏_{P →Q→R} Z(P)

as follows. For each x ∈ N_{G}(P, Q) and y ∈ N_{G}(Q, R),

f_{∗}(c_{y})f_{∗}(c_{x}) = f_{∗}(c_{yx}),

where f_{∗} is as in Lemma 5.3. Hence Φ_{P,R}(yx)^{−1}Φ_{Q,R}(y)Φ_{P,Q}(x) ∈ C_{G′}(P′). Define

ω′_{Φ}(y, x) =^{def} (Φ_{P,R}(yx)^{−1}Φ_{Q,R}(y)Φ_{P,Q}(x))_{p} ∈ Z(P′).

For each P ⟶^{x^{∘}} Q ⟶^{y^{∘}} R in O_{S}^{c}(G), set

ω_{Φ}(y, x) = ω_{Φ}(P ⟶^{x^{∘}} Q ⟶^{y^{∘}} R) =^{def} f^{−1}(ω′_{Φ}(y, x)) ∈ Z(P).

To show that this is well defined we must show that ω′_{Φ}(hy, gx) = ω′_{Φ}(y, x) for any g ∈ Q and any h ∈ R. Indeed

Φ_{P,R}(hygx) = f(hygy^{−1})Φ_{P,R}(yx), Φ_{Q,R}(hy)Φ_{P,Q}(gx) = f(h)Φ_{Q,R}(y)f(g)Φ_{P,Q}(x),

and hence

Φ_{P,R}(hygx)^{−1}Φ_{Q,R}(hy)Φ_{P,Q}(gx)

= Φ_{P,R}(yx)^{−1}f(yg^{−1}y^{−1})Φ_{Q,R}(y)f(g)Φ_{P,Q}(x)

= Φ_{P,R}(yx)^{−1}(f(yg^{−1}y^{−1})Φ_{Q,R}(y)f(g)Φ_{Q,R}(y)^{−1})Φ_{Q,R}(y)Φ_{P,Q}(x).

But f(ygy^{−1}) = Φ_{Q,R}(y)f(g)Φ_{Q,R}(y)^{−1} by (a). Hence

Φ_{P,R}(hygx)^{−1}Φ_{Q,R}(hy)Φ_{P,Q}(gx) = Φ_{P,R}(yx)^{−1}Φ_{Q,R}(y)Φ_{P,Q}(x),

and ω′_{Φ}(hy, gx) = ω′_{Φ}(y, x) as claimed.

To see that ω_{Φ} is a 2-cocycle, we must show, for each sequence

P ⟶^{x^{∘}} Q ⟶^{y^{∘}} R ⟶^{z^{∘}} T

in O_{S}^{c}(G), that

d(ω_{Φ})(P ⟶^{x^{∘}} Q ⟶^{y^{∘}} R ⟶^{z^{∘}} T) =^{def} (x^{−1}ω(z, y)x)·ω(z, yx)^{−1}ω(zy, x)ω(y, x)^{−1} = 1.

For any g ∈ Z(Q),

f(x^{−1}gx) ≡ Φ_{P,R}(yx)^{−1}Φ_{Q,R}(y)f(g)Φ_{Q,R}(y)^{−1}Φ_{P,Q}(yx)

by condition (a) (applied to the squares for y and yx); so it suffices to show that

Φ_{P,R}(yx)^{−1}Φ_{Q,R}(y)f(ω(z, y))Φ_{Q,R}(y)^{−1}Φ_{P,R}(yx)·f(ω(z, yx)^{−1}ω(zy, x)ω(y, x)^{−1}) = 1.

And this follows immediately by substitution and cancellation.

Now set ω(f) = [ω_{Φ}] ∈ \underleftarrow{lim}^{2}_{O_{S}^{c}(G)}(Z). We claim that ω(f) depends only on the iso-

morphism f : S → S′, and that f extends to an isotypical equivalence ψ : L_{S}^{c}(G) → L_{S′}^{c}(G′) of categories (i.e., f_{ψ} = f) if and only if ω(f) = 0. Clearly, the Φ_{P,Q} induce a functor ψ : L_{S}^{c}(G) → L_{S′}^{c}(G′) by taking the corresponding bijections Φ̂_{P,Q} if and only if ω_{Φ} = 0. Also, for any isotypical equivalence ψ : L_{S}^{c}(G) → L_{S′}^{c}(G′) with f_{ψ} = f which sends inclusions in S to inclusions in S′, the maps Φ_{P,Q} defined by ψ between morphism sets satisfy conditions (a) and (b) above by Lemma 5.1(ii). So it remains to show that

1. for any two choices of maps Φ_{P,Q} and Φ′_{P,Q}, the corresponding 2-cycles ω_{Φ} and ω_{Φ′} differ by a coboundary, and 2. all coboundaries can be realized in this way.

To prove point 1, fix maps Φ_{P,Q} and Φ′_{P,Q} satisfying conditions (a), (b), and (c) of

Lemma 5.3. For each morphism P ^{x}→ Q in L_{S}^{c}(G), one has Φ_{P,Q}(x)^{−1}Φ′_{P,Q}(x) ∈ C_{G′}(P′) by condition (a). Set

A(x) =^{def} Φ_{P,Q}(x)^{−1}Φ′_{P,Q}(x); and α(x) =^{def} f^{−1}((A(x))_{p}) ∈ Z(P).  (1)

Then α(x) does not depend of the choice of the lifts Φ_{P,Q} and Φ′_{P,Q}. Also, α(x) depends only on the class x^{∘} ∈ Mor_{O_{S}^{c}(G)}(P, Q) by condition (b). So

α ∈ C^{1}(O_{S}^{c}(G); Z) = ∏_{P →Q} Z(P).

Write a_{x} = α(x) for short. For each sequence P ⟶^{x^{∘}} Q ⟶^{y^{∘}} R in O_{S}^{c}(G), repeated application of condition (a) gives

f(ω_{Φ′}(y, x))

= (Φ′(yx)^{−1}Φ′(y)Φ′(x))_{p}

= (A(yx)^{−1}(Φ(yx)^{−1}Φ(y)Φ(x))(Φ(x)^{−1}A(y)Φ(x))A(x))_{p}

= f(a_{yx})^{−1}ω′_{Φ}(y, x)(Φ(x)^{−1}A(y)Φ(x))_{p}f(a_{x})

= ω′_{Φ}(y, x)[Φ(x)^{−1}f(a_{y})Φ(x)]f(a_{yx})^{−1}f(a_{x})

= ω′_{Φ}(y, x)f((x^{−1}a_{y}x)a_{yx}^{−1}a_{x})

= f(ω_{Φ}(y, x))·f(dα(P ⟶^{x^{∘}} Q ⟶^{y^{∘}} R)).

Thus ω_{Φ′} = ω_{Φ} + dα.

Finally to see 2 above notice that any 1-cochain α can be realized as the difference between Φ and some Φ′, since one can use (1) to define Φ′. This completes the proof of the proposition.

Remark 5.5. Let P_{p}^{c}(G) denote the Frobenius category: the category whose objects are the p-centric subgroups of G, and with Mor_{P_{p}^{c}(G)}(P, Q) = N_{G}(P, Q)/C_{G}(P) (⊆ Hom(P, Q)). Martino and Priddy in [MP] defined an isotypical equivalence of Frobenius categories to be an equivalence ψ : P_{p}^{c}(G) → P_{p}^{c}(G′) together with iso-

morphisms P ^{≅}→ ψ(P) which are natural with respect to all morphisms in P_{p}^{c}(G). They showed [MP, Corollary 1.2] that there is an isotypical equivalence between P_{p}^{c}(G) and P_{p}^{c}(G′) if and only if there is a fusion preserving isomorphism between the Sylow p-subgroups of G and G′. Stewart Priddy pointed out to us that the obstruction for lifting an isotypical equivalence of Frobenius categories to an equivalence of categories L_{p}^{c}(−) lies in \underleftarrow{lim}^{2}_{P_{p}^{c}(G)}(Z), in contrast to the obstruction group of Proposition 5.4. There is a homomorphism

\underleftarrow{lim}^{2}_{O_{p}^{c}(G)}(Z) ⟶ \underleftarrow{lim}^{2}_{P_{p}^{c}(G)}(Z)

which sends our obstruction to theirs, but it seems quite possible that our obstruction could be nonzero and theirs zero. The difference between them is that ω(f) ∈ \underleftarrow{lim}^{2}_{O_{p}^{c}(G)}(Z) is the obstruction to constructing an equivalence ψ such that f_{ψ} = f,

while its image in \underleftarrow{lim}^{2}_{P_{p}^{c}(G)}(Z) is the obstruction to constructing ψ such that f_{ψ} ≡ f

modulo Z(S).

Given a fusion preserving isomorphism f between Sylow p-subgroups of G and G′, one can try directly to construct a homotopy equivalence

BG_{p}^{∧} ≃ (\underrightarrow{hocolim}_{P ∈O_{p}^{c}(G)}(EG/P))_{p}^{∧} ⟶ BG′_{p}^{∧}.

Maps EG/P → BG′_{p}^{∧} are determined up to homotopy by f, and the obstruction to extending them to a map defined on the homotopy colimit lies precisely in \underleftarrow{lim}^{2}_{O_{p}^{c}(G)}(Z).

It was this observation — that one finds the same obstruction group to constructing an equivalence of categories L_{p}^{c}(−) and to constructing an equivalence of spaces — which first suggested Theorem A to us.

Just as the obstruction to realizing a fusion preserving isomorphism as the underlying isomorphism of an equivalence of categories L_{p}^{c}(−) lies in \underleftarrow{lim}^{2}(Z), the obstruction to the uniqueness of such a realization lies in \underleftarrow{lim}^{1}(Z). To simplify this discussion, we restrict attention to automorphisms of L_{p}^{c}(G). Recall that Out_{typ}(L_{p}^{c}(G)) denotes the group of isotypical self equivalences modulo natural isomorphism. By analogy, we let Aut_{fus}(S), for any Sylow p-subgroup S ≤ G, denote the group of fusion preserving automorphisms of S, and set

Out_{fus}(S) = Aut_{fus}(S)/{c_{x} | x ∈ N_{G}(S)}.

We are now ready to restate and prove Theorem E. As pointed out in the introduction, this is an algebraic version of the exact sequence of [BL, Theorem 1.6], where Out_{typ}(L_{p}^{c}(G)) has been replaced by Out(BG_{p}^{∧}).

Theorem 5.6. For any finite group G, there is an exact sequence

0 ⟶ \underleftarrow{lim}^{1}_{O_{p}^{c}(G)}(Z) ⟶^{λ_{G}} Out_{typ}(L_{p}^{c}(G)) ⟶^{μ_{G}} Out_{fus}(S) ⟶^{ω_{G}} \underleftarrow{lim}^{2}_{O_{p}^{c}(G)}(Z),

where λ_{G} and μ_{G} are group homomorphisms.

Proof. As in the proof of Proposition 5.4, we replace L_{p}^{c}(G) and O_{p}^{c}(G) by their equivalent subcategories L_{S}^{c}(G) and O_{S}^{c}(G): the full subcategories whose objects are the subgroups of S which are p-centric in G.

Let ω_{G} be the obstruction map of Proposition 5.4, and let μ_{G} be the map which sends ψ ∈ Aut_{typ}(L_{p}^{c}(G)) to f_{ψ}. If ψ and ψ′ are naturally isomorphic, by a natural isomorphism which sends S to x̂ ∈ Aut_{L_{S}^{c}(G)}(S), then f_{ψ} and f_{ψ′} differ by conjugation by x ∈ N_{G}(S). So μ_{G} is well defined, and the sequence is exact at Out_{fus}(S) by Proposition 5.4.

A 1-cocycle α ∈ Z^{1}(O_{S}^{c}(G); Z) is a collection of maps Mor_{O_{S}^{c}(G)}(P, Q) ^{α_{P,Q}} → Z(P) such that

(x^{−1}α(y)x)·α(yx)^{−1}·α(x) = 1  (1)

for all (P ^{x^{∘}}→ Q ^{y^{∘}}→ R). In particular, α(g^{∘}) = α(Id_{P}) = 1 for all g ∈ P. For any such α, set λ_{G}(α) = [ψ_{α}], where ψ_{α}(P) = P for all objects P, and ψ_{α}(x̂) = { \atop xα(x)}. Then ψ_{α} is a functor by (1). And if β is a 0-cochain, then the elements β(P) ∈ Z(P) define a natural isomorphism Id → ψ_{dβ}. One easily checks that ψ_{α+α′} = ψ_{α} ∘ ψ′_{α}; and this shows that λ_{G} is a well defined homomorphism.

It is straightforward to show that Im(λ_{G}) = Ker(μ_{G}). To see that λ_{G} is injective, fix [α] ∈ Ker(λ_{G}), and let β : Id → ψ_{α} be a natural isomorphism. This consists of a choice of { \atop β(P)} ∈ Aut_{L_{p}^{c}(G)}(P) for each P, where β(P) ∈ N(P). Naturality with

respect to automorphisms P ĝ→ P for g ∈ P implies that β(P) ∈ Z(P) for each P (since ψ_{α}(ĝ) = ĝ). Thus β ∈ C^{0}(O_{S}^{c}(G); Z), and α = dβ.

The next proposition deals with the question whether or not the obstruction map is a homomorphism. As we are not aware of an example of a finite group G, where the corresponding \underleftarrow{lim}^{2}(Z) is non-zero, we obviously don’t have an example where ω_{G} fails to be a homomorphism. However, the proposition shows that ω_{G} should be thought of as a 1-cocycle rather than a homomorphism.

Notice that Out_{fus}(S) acts on \underleftarrow{lim}^{2}(Z) on the right. The action is given as follows. Let f ∈ Aut_{fus}(S) be a fusion preserving automorphism of S. The group C^{2}(O_{S}^{c}(G); Z) is generated by functions ν from the collection of all 2-chains of morph-

isms P ^{x^{∘}}→ Q ^{y^{∘}}→ R in O_{p}^{c}(G) to the product of all centers of p-centric subgroups of S, where ν(y, x) ∈ Z(P). Choose maps Φ_{P,Q} as in Lemma 5.3, and define

(ν · f)(y, x) = f^{−1}(ν(Φ_{Q,R}(y), Φ_{P,Q}(x))).

We leave it to the reader to verify that this is well defined and gives the required right action.

Proposition 5.7. For any finite group G, the obstruction map

ω_{G} : Out_{fus}(S) ⟶ \underleftarrow{lim}^{2}_{O_{p}^{c}(G)}(Z)

satisfies the relation

ω_{G}(f_{2} ∘ f_{1}) = ω_{G}(f_{1}) + (ω_{G}(f_{2}))f_{1}.

In particular ω_{G} is a homomorphism if the action of Out_{fus}(S) on \underleftarrow{lim}^{2}(Z) is trivial.

Proof. For any p-subgroups P, Q ≤ G, any x ∈ N_{G}(P, Q), and any g ∈ P,

Φ_{P,Q}(xg) = f(xgx^{−1})·Φ_{P,Q}(x) = Φ_{P,Q}(x)·f(g) :

the first equality follows from (b) and the second from (a). Hence for any x, x′ ∈ N_{G}(P, Q) such that x^{−1}x′ ∈ P,

(Φ_{P,Q}(x))^{−1}·Φ_{P,Q}(x′) = f(x^{−1}x′).  (1)

Now fix fusion preserving automorphisms f_{1}, f_{2} of S. For each i = 1, 2, let Φ_{P,Q}^{i} be maps which satisfy conditions (a), (b), and (c) of Lemma 5.3 relative to f_{i}. To simplify the notation in what follows, we drop the subscripts whenever possible. Fix

a sequence of morphisms P ^{x^{∘}}→ Q ^{y^{∘}}→ R in O_{p}^{c}(G). Then

(f_{2} ∘ f_{1})(ω(f_{2} ∘ f_{1})(y, x)) = ((Φ^{2}Φ^{1}(yx))^{−1}·Φ^{2}Φ^{1}(y)·Φ^{2}Φ^{1}(x))_{p}
= ((Φ^{2}Φ^{1}(yx))^{−1}·(Φ^{2}(Φ^{1}(y)Φ^{1}(x))))_{p}
·((Φ^{2}(Φ^{1}(y)Φ^{1}(x)))^{−1}·Φ^{2}Φ^{1}(y)·Φ^{2}Φ^{1}(x))_{p}.

By (1),

((Φ^{2}Φ^{1}(yx))^{−1}·(Φ^{2}(Φ^{1}(y)Φ^{1}(x))))_{p} = f_{2}((Φ^{1}(yx)^{−1}·Φ^{1}(y)Φ^{1}(x))_{p});

and by definition

((Φ^{2}(Φ^{1}(y)Φ^{1}(x)))^{−1}·Φ^{2}Φ^{1}(y)·Φ^{2}Φ^{1}(x))_{p} = f_{2}(ω(f_{2})(Φ^{1}(y), Φ^{1}(x))).

Hence

ω(f_{2} ∘ f_{1})(y, x) = ω(f_{1})(y, x)·f_{1}^{−1}(ω(f_{2})(Φ^{1}(y), Φ^{1}(x))),

and so

ω(f_{2} ∘ f_{1}) = ω(f_{1}) + (ω(f_{2}))f_{1}.

We now discuss vanishing of higher limits groups over orbit categories. Recall that the p-rank rk_{p}(G) of a finite group G is the rank of the largest elementary abelian p-subgroup of G. In particular, we show that \underleftarrow{lim}^{r}_{O_{p}^{c}(G)}(Z) = 0 whenever rk_{p}(G) < p^{r}.

One way to compute higher limits of functors over orbit categories is to filter the functor in a way so that each of the quotient functors vanishes except on one isomorphism class of objects. In particular, if higher limits of the original functor are non-vanishing, then they must also be non-vanishing for at least one of the quotient functors. This motivated the definition of graded abelian groups Λ^{∗}(G; M) in [JMO]. For any prime p, any finite group G, and any ℤ_{(p)}[G]-module M,

Λ^{∗}(G; M) =^{def} \underleftarrow{lim}^{∗}_{O_{p}(G)}(F_{M}), where F_{M}(P) = \begin{cases}M & if P = 1 \\ 0 & otherwise. \end{cases}

These groups have the property [JMO, Lemma 5.4] that if F : O_{p}(G)^{op} ⟶ ℤ_{(p)}-mod is any functor which vanishes except on subgroups conjugate to P, then \underleftarrow{lim}^{∗}(F) ≅ Λ^{∗}(N(P)/P; F (P)).

Proposition 5.8. For any finite group G, any finitely generated 𝔽_{p}[G]-module M, and any k ≥ 1 such that Λ^{k}(G; M) ≠ 0, there is an elementary abelian p-subgroup A ≤ G of rank k such that M|_{A} contains the free module 𝔽_{p}[A] as a direct summand. In particular,

rk_{p}(G) ≥ k  and  dim_{𝔽_{p}}(M) ≥ p^{k}.

Proof. Let S_{p}(G) denote the nerve of the poset of nontrivial p-subgroups of G, and let C̃_{∗}(S_{p}(G)) denote its reduced chain complex. Thus, C̃_{−1}(S_{p}(G)) = ℤ, and C̃_{k}(S_{p}(G)) (for k ≥ 0) is the free ℤ-module with basis the set of sequences 1 ≠ P_{0} ≨ P_{1} ≨ ⋯ ≨ P_{k}. The conjugation action of G on the basis makes this into a chain complex of ℤ[G]-modules. By [Gr, Theorem 1.2], for any ℤ_{(p)}[G]-module M,

Λ^{∗}(G; M) ≅ H^{∗−1}(Hom_{G}(C̃_{∗}(S_{p}(G)), M)).

Similarly, let A_{p}(G) denote the nerve of the poset of nontrivial elementary abelian p-subgroups of G, and set C_{∗} = C̃_{∗}(A_{p}(G); 𝔽_{p}). By [Qu], A_{p}(G) and S_{p}(G) are Ghomotopy equivalent, and so Λ^{∗}(G; M) ≅ H^{∗−1}(Hom_{G}(C_{∗}, M)) for any 𝔽_{p}[G]-module M. By [Wb2, Theorem 2.7.1], C_{∗} splits as a sum C_{∗} = D_{∗} ⊕P_{∗} of complexes of 𝔽_{p}[G]- modules, where D_{∗} is exact (in fact, split exact), and where each P_{k} is projective. In particular, Λ^{∗}(G; M) is the cohomology of the complex Hom_{G}(P_{∗}, M).

Assume that Λ^{k}(G; M) ≠ 0. Then Hom_{G}(P_{k−1}, M) ≠ 0 by the above remarks. Since P_{k−1} is 𝔽_{p}[G]-projective, the module Hom_{𝔽_{p}}(P_{k−1}, M) is also 𝔽_{p}[G]-projective. Notice that for any projective G-module P, the submodule of G-invariants is given by

P^{G} = {ν_{G} · x | x ∈ P },

where ν_{G} is the norm element ∑_{g∈G} g ∈ 𝔽_{p}[G]. The group G acts on Hom_{𝔽_{p}}(P_{k−1}, M) by (gϕ)(x) = g · ϕ(g^{−1}x), and

0 ≠ Hom_{G}(P_{k−1}, M) ≅ (Hom_{𝔽_{p}}(P_{k−1}, M))^{G}.

Hence there is some ϕ : P_{k−1} → M such that ∑_{g∈G} gϕ ≠ 0. By extension, we may assume that ϕ with this property is defined on C_{k−1} = D_{k−1} ⊕ P_{k−1}.

The complex C_{∗} is generated as an 𝔽_{p}[G]-module in dimension k −1 by the (k −1)- simplices of A_{p}(G), namely, by sequences of the form τ = (E_{1} ≨ ⋯ ≨ E_{k}). Hence there exists such a simplex τ such that

∑_{g∈G}(gϕ)(τ) = ∑_{g∈G} g · ϕ(g^{−1}τ) ≠ 0,

where ϕ is as above.

Let b_{1}, …, b_{r} be a choice of right coset representatives for E_{k} in G. Notice that E_{k} is contained in the stabilizer group of τ. Thus

0 ≠ ∑_{g∈G}(gϕ)(τ) = ∑_{g∈E_{k}} ∑_{i=1}^{r} gb_{i}ϕ(b_{i}^{−1}τ).

Set x = ∑_{i=1}^{r} b_{i}ϕ(b_{i}^{−1}τ) ∈ M. Then ∑_{g∈E_{k}} gx ≠ 0. Notice that rk_{p}(G) ≥ rk_{p}(E_{k}) ≥ k.

Let ι_{x} : 𝔽_{p}[E_{k}] → M denote the E_{k}-map sending 1 to x. Then Ker(ι_{x}) does not contain the norm element ν_{E_{k}} and hence is the zero ideal in the group ring (cf. [Se, §8.3, Prop. 26]). Thus ι_{x} is a monomorphism. Furthermore, since 𝔽_{p}[E_{k}] is an injective E_{k}-module, the module M|_{E_{k}} contains 𝔽_{p}[E_{k}] as a split summand. In particular, dim_{𝔽_{p}}(M) ≥ |E_{k}| ≥ p^{k}. This completes the proof.

The idea of the last part of the above proof was suggested to us by the proof of [Wb1, 5.3].

The inequalities of Proposition 5.8 are in fact the best possible, as is shown by the following examples. Let V denote the 𝔽_{p}[Σ_{p+1}]-module V = (𝔽_{p})^{p+1}/(diag). Then Λ^{1}(Σ_{p+1}; V) ≅ 𝔽_{p} by [JMO, Proposition 6.2(i)]. So for any k > 0,

Λ^{k}((Σ_{p+1})^{k}, V^{⊗k}) ≠ 0

by the Künneth formula in [JMO, Proposition 6.1(v)]; and

Λ^{k}(Σ_{p+1} \underbrace{≀C_{p} ≀ ⋯ ≀ C_{p}}_{(k − 1) times} , V^{p^{k−1}})≠ 0

by a general formula Λ^{i}(G ≀ C_{p}; M^{p}) ≅ Λ^{i−1}(G; M) when p||G| (unpublished). These are thus two examples of pairs (G, M) for which Λ^{k}(G; M) ≠ 0, rk_{p}(G) = k, and dim_{𝔽_{p}}(M) = p^{k}.

When k = 1, a similar argument gives a much stronger result. If M is an 𝔽_{p}[G]- module with Λ^{1}(G; M) ≠ 0, then for any Sylow p-subgroup S ≤ G, M|_{S} contains a summand isomorphic to 𝔽_{p}[S], and in particular dim_{𝔽_{p}}(M) ≥ |S|.

As an easy consequence of Proposition 5.8, we get:

Corollary 5.9. Fix a finite group G, and a functor F : O_{p}^{c}(G)^{op} → p-Gr. Assume, for some k > 0, that \underleftarrow{lim}^{k}_{O_{p}^{c}(G)}(F) ≠ 0. Then there is a p-centric subgroup P ≤ G such

that rk_{p}(F (P)) ≥ p^{k}.

Proof. Extend F to a functor on O_{p}(G) by setting F (P) = 0 whenever P is not p-centric. Since any p-subgroup of G which contains a p-centric subgroup is itself p-centric, the chain complexes C^{∗}(O_{p}(G); F) and C^{∗}(O_{p}^{c}(G); F) are isomorphic. So higher limits of F are the same whether taken over O_{p}(G) or over O_{p}^{c}(G).

Now let P_{1}, P_{2}, …, P_{k} be conjugacy class representatives for p-centric subgroups of G, arranged such that |P_{1}| ≤ |P_{2}| ≤ · · ·. For 0 ≤ i ≤ k, let F_{i} be the functor

F_{i}(P) = \begin{cases}0 & if P ≥ xP_{j}x^{−1}, some x ∈ G and j > i \\ F (P) & otherwise. \end{cases}

Thus, F_{0} = 0, F_{k} = F, and F_{i} is a subfunctor of F_{i+1} for each i. By the exact sequences of higher limits for extensions of functors, there is some 1 ≤ i ≤ k such that \underleftarrow{lim}^{k}(F_{i}/F_{i−1}) ≠ 0. The functor F_{i}/F_{i−1} vanishes except on subgroups conjugate to P_{i}, and hence

Λ^{k}(N(P_{i})/P_{i}; F (P_{i})) ≅ \underleftarrow{lim}^{k}_{O_{p}(G)}(F_{i}/F_{i−1}) ≠ 0

by [JMO, Lemma 5.4]. Then Λ^{k}(N(P)/P; p^{j}F (P)/p^{j+1}F (P)) ≠ 0 for some j ≥ 0 (where P = P_{i}), so this module has rank at least p^{k} by Proposition 5.8, and hence rk_{p}(F (P)) ≥ p^{k}.

We are now ready to prove the first half of Theorem F.

Theorem 5.10. Fix a prime p, and a finite group G such that rk_{p}(G) < p^{2}. Then for any other finite group G′, BG_{p}^{∧} ≃ BG′_{p}^{∧} if and only if there is a fusion preserving isomorphism between Sylow p-subgroups of G and of G′.

Proof. If BG_{p}^{∧} ≃ BG′_{p}^{∧}, then there is an equivalence of categories

ψ : L_{p}^{c}(G) ⟶^{≃} L_{p}^{c}(G′)

by Theorem A, and hence a fusion preserving isomorphism between the Sylow psubgroups by Lemma 5.1.

Conversely, if rk_{p}(G) < p^{2} then rk_{p}(Z(P)) < p^{2} for each p-centric P ≤ G, and Corollary 5.9 implies that \underleftarrow{lim}^{2}(Z) = 0. Here, as usual, Z : O_{p}^{c}(G) → p-Gr is the functor which sends P to Z(P). So if there is a fusion preserving isomorphism between Sylow p-subgroups of G and G′, then the categories L_{p}^{c}(G) and L_{p}^{c}(G′) are equivalent by Proposition 5.4, and so BG_{p}^{∧} ≃ BG′_{p}^{∧} by Theorem A.

6. A direct construction of equivalences between categories

We now describe another, more direct approach to constructing equivalences of categories L_{p}^{c}(−). The following lemma will be useful when constructing functors. It follows immediately from the definition of the categories L_{p}^{c}(−).

Lemma 6.1. For any finite group G all morphisms in L_{p}^{c}(G) are monomorphism in the categorical sense. Namely, for any p-centric subgroups P, Q, R ≤ G, and any x̂, ŷ ∈ Mor_{L_{p}^{c}(G)}(P, Q) and ẑ ∈ Mor_{L_{p}^{c}(G)}(Q, R), ẑ ∘ x̂ = ẑ ∘ ŷ implies x̂ = ŷ.

A p-subgroup P ≤ G will be called p-radical if O_{p}(N_{G}(P)) = P; i.e., if N_{G}(P)/P contains no nontrivial normal p-subgroups. In [JMO] such subgroups are called pstubborn. We let L_{p}^{cr}(G) ⊆ L_{p}^{c}(G) denote the full subcategory whose objects are the p-radical p-centric subgroups of G. Also, for any Sylow p-subgroup S ≤ G, we let

L_{S}(G) ⊆ L_{p}(G),  L_{S}^{c}(G) ⊆ L_{p}^{c}(G),  and L_{S}^{cr}(G) ⊆ L_{p}^{cr}(G)

denote the full subcategories whose objects in each case are those subgroups in the larger category which are contained in S. All three of these inclusions are equivalences of categories.

The following lemma shows that equivalences between categories L_{S}^{cr}(−) are essentially the same as equivalences between the L_{S}^{c}(−).

Lemma 6.2. Fix finite groups G and G′, and Sylow p-subgroups S ≤ G and S′ ≤ G′. Let ψ_{0} : L_{S}^{cr}(G) → L_{S′}^{cr}(G′) be an isotypical equivalence of categories which sends inclusions to inclusions. Then ψ_{0} extends to a unique isotypical equivalence ψ : L_{S}^{c}(G) → L_{S′}^{c}(G′) which sends inclusions to inclusions.

Proof. The equivalence ψ_{0} will be extended to an equivalence ψ as above in several steps. To keep the notation simple, these equivalences will all be denoted ψ. So we start by setting ψ = ψ_{0}: an equivalence between the full subcategories of p-centric p-radical subgroups.

For each P ≤ S, set P = O_{p}(N_{G}(P)) (the intersection of the Sylow p-subgroups of N_{G}(P)) if P is p-centric, and let P be a Sylow p-subgroup of C_{G}(P)·P otherwise. For each P ≤ S not in L_{S}^{cr}(G), P satisfies the two conditions: P ≨ P ≤ N_{G}(P), and N_{G}(P) = (N_{G}(P) ∩ N_{G}(P))·C_{G}(P). Since S is p-centric and p-radical, ψ is already defined there, and the argument in the proof of Lemma 5.1 now carries through to show that ψ_{S,S′} restricts to a fusion preserving isomorphism

f = f_{ψ} = ψ_{S} : S ⟶^{≅} S′.

Also, for any p-centric p-radical subgroups P, Q ≤ S, and any x ∈ N_{G}(P, Q) and x′ ∈ N_{G′}(f(P), f(Q)) such that ψ_{P,Q}(x̂) = x̂′, the following square commutes by the same argument as in the proof of Lemma 5.3:

\xymatrix{P \ar[r]^{c_{x}} \ar[d]_{f} & Q \ar[d]_{f} \\ f(P) \ar[r]^{c_{x′}} & f(Q).}  (1)

By Lemma 5.2, for any P ≤ S, P is p-centric in G if and only if f(P) is p-centric in G′. So we can define ψ : L_{S}^{c}(G) → L_{S′}^{c}(G′) on objects by setting ψ(P) = f(P) for all p-centric P ≤ S. It remains to define ψ on morphisms in L_{S}^{c}(G).

Since we will be sending inclusions to inclusions (and every morphism is the composite of an isomorphism followed by an inclusion), the problem is to define ψ on isomorphisms. The basic idea is to define ψ_{P,Q}(x̂), for any isomorphism P x̂→ Q in L_{S}^{c}(G) for which P and Q are not p-radical, to be the “restriction” of ψ_{P ,Q}(x̂); i.e., the morphism (unique by Lemma 6.1) such that ˆ1 ∘ ψ_{P,Q}(x̂) = ψ_{P ,Q}(x̂) ∘ ˆ1. Here,

P = O_{p}(N_{G}(P)) for p-centric P ≤ S as above. However, since P need not be p-radical (when P is not p-radical), we must combine this with a downward induction on |P |. And the procedure is further complicated since for P ≤ S not p-radical, P need not be contained in S if N_{S}(P) is not a Sylow p-subgroup of N_{G}(P).

Set p^{k} = |S|. For each 0 ≤ i ≤ k, let L_{i} ⊆ L_{S}^{c}(G) be the full subcategory whose objects are the p-centric subgroups P ≤ S which are either p-radical in G, or which satisfy |P | ≥ p^{i}. In particular, L_{k} = L_{S}^{cr}(G) and L_{0} = L_{S}^{c}(G). Fix 0 ≤ i < k, and assume inductively that ψ has already been defined on L_{i+1}, and that (1) commutes whenever P and Q are in L_{i+1}. We must extend this to L_{i}, and so that (1) still commutes when appropriate.

Set P_{i} = Ob(L_{i})∖ Ob(L_{i+1}), the set of p-centric subgroups P ≤ S of order p^{i} which are not p-radical. Let P_{i}^{∗} ⊆ P_{i} be a set of G-conjugacy class representatives, chosen such that for each P^{∗} ∈ P_{i}^{∗}, N_{S}(P^{∗}) is a Sylow p-subgroup of N_{G}(P^{∗}). For each P ∈ P_{i}, let P^{∗} denote the unique element of P_{i}^{∗} which is G-conjugate to P, and fix some element x_{P} ∈ N_{G}(P, P^{∗}) such that x_{P} N_{S}(P)x_{P}^{−1} ≤ N_{S}(P^{∗}).

Fix some P ∈ P_{i} and some x ∈ N_{G}(P^{∗}). We claim that there are unique morphisms ψ_{P,P^{∗}}(x̂_{P}) and ψ_{P^{∗},P^{∗}}(x̂) such that the following squares commute:

\xymatrix{f(P) \ar[rr]^{ψ_{P,P^{∗}}(x̂_{P})} \ar[dd]_{ˆ1} &  & f(P^{∗}) \ar[dd]_{ˆ1} &  & f(P^{∗}) \ar[rr]^{ψ_{P^{∗},P^{∗}}(x̂)} \ar[dd]_{ˆ1} &  & f(P^{∗}) \ar[dd]_{ˆ1} \\  & (2) &  & and &  & (3) \\ f(N_{S}(P)) \ar[rr]^{ψ_{N_{S} (P),N_{S} (P^{∗})}(x̂_{P})} &  & f(N_{S}(P^{∗})) &  & f(P^{∗}) \ar[rr]^{ψ_{P^{∗},P^{∗}}(x̂)} &  & f(P^{∗}).}

To see this, set P̂ = N_{S}(P) and y = x_{P} if P ≠ P^{∗}, and P̂ = P and y = x if P = P^{∗}. In either case, P̂ ≩ P (so P̂ is in L_{i+1}) and y ∈ N_{G}(P̂, N_{S}(P^{∗})). Choose y′ ∈ N_{G′}(f(P̂), f(N_{S}(P^{∗}))) such that ψ_{P̂,N_{S}(P^{∗})}(ŷ) = ŷ′. Then y′ ∈ N_{G′}(f(P), f(yP y^{−1})) by (1). So if we set ψ_{P,P^{∗}}(ŷ) = ŷ′, then the above square (2) or (3) commutes, and the choice of ŷ′ to make this commute is unique by Lemma 6.1.

If P x̂→ Q is an arbitrary isomorphism between objects of L_{i} not in L_{i+1}, then it factors uniquely as a composite

P \underrightarrow{x̂_{P}} P^{∗} \underrightarrow{ŷ} P^{∗} = Q^{∗} ⟶^{x̂_{Q}^{−1}} Q  (4)

(P^{∗} = Q^{∗} since P and Q are conjugate); and we set ψ_{P,Q}(x̂) =^{def} x̂′_{Q}^{−1} ∘ ψ_{P^{∗},P^{∗}}(ŷ) ∘ x̂′_{P}.

And if P x̂→ Q is an arbitrary morphism in L_{i}, then it factors as the composite of the isomorphism P \underrightarrow{x̂} ^{x}P (recall that ^{x}P = xP x^{−1}) followed by the inclusion

xP ˆ1→ Q, and we set ψ_{P,Q}(x̂) =^{def} ˆ1 ∘ ψ_{P,^{x}P} (x̂).

We have now defined ψ on all morphisms of L_{i}, and it remains to show that it preserves composition of morphisms. This means showing, for each P x̂→ Q ŷ→ R in L_{i}, that the following diagram commutes:

[figure: f(P); ψ_{P,^{yx}P} (ŷx); ψ_{P,^{x}P} (x̂); (5); f(^{yx}P); ˆ1; ˆ1; f(R).; (7); f(^{x}P); ψ_{^{x}P,^{yx}P} (ŷ); ˆ1; (6); f(^{y}Q); ˆ1; f(Q); ψ_{Q,^{y}Q}(ŷ)]

Square (7) clearly commutes. Via the decomposition (4) it suffices to show the commutativity of (5) when P = ^{x}P = ^{yx}P = P^{∗}, and this follows from the uniqueness of the morphisms ψ_{P,P} (−) which make square (3) commute. So it remains only to show that (6) commutes. It suffices to show this when either P or Q lies in P_{i}, but not both (if P, Q ∈ P_{i} then ^{x}P = Q since both have order p^{i}).

We are thus reduced to showing that the following two squares commute

\xymatrix{f(Q) \ar[r]^{ˆ1} \ar[d]_{ψ_{Q,^{x}Q}(x̂)}^{≅} & f(P) \ar[d]_{ψ_{P,^{x}P} (x̂)}^{≅} & f(P) \ar[r]^{ˆ1} \ar[d]_{ψ_{P,^{x}P} (x̂)}^{≅} & f(R) \ar[d]_{ψ_{R,^{x}R}(x̂)}^{≅} \\ f(^{x}Q) \ar[r]^{ˆ1} & f(^{x}P) & f(^{x}P) \ar[r]^{ˆ1} & f(^{x}R)}

for any Q ≤ P ≤ R ≤ S and any x ∈ G such that P ∈ P_{i}, Q and R are in L_{i+1}, and ^{x}R ≤ S. Since P x̂→^{x}P factors as in (4), it suffices to do this when ^{x}P = P^{∗}, and either P = P^{∗} ∈ P_{i}^{∗} or x = x_{P}. If x = x_{P}, then this follows from the following diagram:

\xymatrix{f(Q) \ar[r]^{ˆ1} \ar[d]_{ψ_{Q,^{x}Q}(x̂_{P})} & f(P) \ar[r]^{ˆ1} \ar[d]_{ψ_{P,^{x}P} (x̂_{P})} & f(R) \ar[r]^{ˆ1} \ar[d]^{ψ_{R,^{x}R}(x̂_{P})} & f(T) \ar[d]^{ψ_{T,^{x}T} (x̂_{P})} \\ f(^{x}Q) \ar[r]^{ˆ1} & f(^{x}P) \ar[r]^{ˆ1} & f(^{x}R) \ar[r]^{ˆ1} & f(^{x}T),}

where T = R·N_{S}(P). The composite of the two squares on the right commutes by (2). If Q ≨ P, then the composite of the three squares commutes since it is in L_{i+1}; otherwise the first square commutes trivially. Similarly, the third square commutes if P ≨ R and the second square otherwise. With the help of Lemma 6.1, this shows that all three squares commute (in all cases). A similar argument applies when P = ^{x}P ∈ P_{i}^{∗}, upon using (3) and setting T = R·P.

We have now constructed a functor ψ : L_{S}^{c}(G) → L_{S′}^{c}(G′). From the above procedure (and Lemma 6.1) this is the unique such functor which extends ψ_{0} and sends inclusions to inclusions. Upon applying this same procedure to (ψ_{0})^{−1}, we obtain a functor ψ′ in the opposite direction; the two composites ψ ∘ ψ′ and ψ′ ∘ ψ are the identity by the uniqueness of such extensions; and this shows that ψ is an isomorphism of categories.

The following proposition is our main tool for directly (without reference to higher limits) constructing equivalences between the categories L_{p}^{c}(−). To simplify the notation, for a p-centric subgroup P ≤ G, we write N′(P) = N_{G}(P)/C′_{G}(P), where C′_{G}(P) is the p′-torsion in the centralizer of P. (Note that N′(P) is the automorphism group of P in the category L_{p}^{c}(G).) And we let Equiv_{typ}(L_{p}^{c}(G), L_{p}^{c}(G′)) denote the set of isotypical equivalences between these categories modulo natural isomorphisms of functors.

Fix finite groups G and G′ and Sylow p-subgroups S ≤ G and S′ ≤ G′. Let S = P_{0}, P_{1}, …, P_{m} denote a choice of G-conjugacy class representatives for all pcentric p-radical subgroups of G contained in S. Let X(G, G′) be the set of all (m + 1)-tuples (θ; θ_{1}, θ_{2}, …, θ_{m}) such that

θ : N′(S) ⟶^{≅} N′(S′) and θ_{i} : N′(P_{i}) ⟶^{≅} N′(θP_{i})

are isomorphisms, and such that

θ_{i}|_{N_{G}(S)∩N_{G}(P_{i})} = θ|_{N_{G}(S)∩N_{G}(P_{i})}

for each i. (Here, by restriction to N_{G}(S) ∩ N_{G}(P_{i}), we really mean restriction to its image in N′(P_{i})). The group N_{G′}(S′) act on X(G, G′) by

x · (θ; θ_{1}, …, θ_{m}) =^{def} (c_{x} ∘ θ; c_{x} ∘ θ_{1}, …, c_{x} ∘ θ_{m})

for x ∈ N_{G′}(S′).

Define a map

Λ : Equiv_{typ}(L_{p}^{c}(G), L_{p}^{c}(G′)) ⟶ X(G, G′)/N_{G′}(S′)

as follows. For any isotypical equivalence ψ : L_{p}^{c}(G) → L_{p}^{c}(G′), let ψ′ be an isotypical equivalence which sends inclusions of subgroups of S to inclusions of subgroups of S′ and is naturally isomorphic to ψ. Define

Λ([ψ]) = [ψ_{S,S}; ψ_{P_{1},P_{1}}, …, ψ_{P_{m},P_{m}}],

where ψ_{P,P} : N′(P) → N′(ψ(P)) for P = S or P = P_{i}.

To see that Λ is well defined, replace, as usual, the categories L_{p}^{c}(G) and L_{p}^{c}(G′) by their equivalent subcategories L_{S}^{c}(G) and L_{S′}^{c}(G′). By Lemma 5.1(i,ii), any isotypical equivalence L_{S}^{c}(G) → L_{S′}^{c}(G′) is naturally isomorphic to an isotypical equivalence ψ which sends inclusions to inclusions, and ψ(P) = f_{ψ}(P) for all P ≤ S. So

(ψ_{S,S}; ψ_{P_{1},P_{1}}, …, ψ_{P_{m},P_{m}}) ∈ X(G, G′).

If ψ′ is another isotypical equivalence which sends inclusions to inclusions, and if α : ψ → ψ′ is a natural isomorphism, then set α(S) = x̂ and α(P_{i}) = x̂_{i} for each i. Here, x ∈ N_{G′}(S′), and x_{i} ∈ N_{G′}(ψ(P_{i}), ψ′(P_{i})). The naturality of α implies that the following squares commute for each 0 ≤ i ≤ m and each g ∈ N_{G}(P_{i}):

\begin{matrix}ψ(P_{i}) & \overline{\underrightarrow{ˆ1}} & S′ \\ ψ′(P_{i}) & x̂_{i} ; ↓ & \overline{\underrightarrow{ˆ1}} & S′ & x̂ ; ↓ \end{matrix}  \begin{matrix}ψ(P_{i}) & \overline{→^{ψ_{P_{i},P_{i}}(ĝ)}} & ψ(P_{i}) \\ ψ′(P_{i}) & x̂_{i} ; ↓ & \overline{→^{ψ′_{P_{i},P_{i}}(ĝ)}} & ψ′(P_{i}). & x̂_{i} ; ↓ \end{matrix}

This shows first that x_{i} ≡ x (mod C′_{G′}(P′_{i})) for each i, and that ψ′_{P_{i},P_{i}} = c_{x_{i}} ∘ ψ_{P_{i},P_{i}} = c_{x} ∘ ψ_{P_{i},P_{i}}. The (m + 1)-tuples (ψ_{S,S}; ψ_{P_{1},P_{1}}, …, ψ_{P_{m},P_{m}}) and (ψ′_{S,S}; ψ′_{P_{1},P_{1}}, …, ψ′_{P_{m},P_{m}}) are thus in the same N_{G′}(S′)-orbit of X(G, G′), and this shows that Λ is well defined.

Proposition 6.3. The map

Λ : Equiv_{typ}(L_{p}^{c}(G), L_{p}^{c}(G′)) ⟶ X(G, G′)/N_{G′}(S′)

is injective. Furthermore, Λ is a bijection if there is no 1 ≤ i, j ≤ m such that P_{i} is conjugate to a proper subgroup of P_{j}.

Proof. Proof of injectivity: Fix isotypical equivalences ψ and ψ′ such that Λ([ψ]) = Λ([ψ′]). We can assume that ψ and ψ′ both send inclusions of subgroups of S to inclusions of subgroups of S′; and also (since the action of any x ∈ N_{G′}(S′) can be lifted to a natural isomorphism of equivalences) that ψ_{S,S} = ψ′_{S,S} and ψ_{P_{i},P_{i}} = ψ′_{P_{i},P_{i}} for each i.

Since each morphism factors as an isomorphism followed by an inclusion, it suffices to show that ψ = ψ′ on isomorphisms. So fix a G-conjugacy class P of subgroups of S, and assume inductively that ψ = ψ′ on all isomorphisms of subgroups of strictly larger order. For any P, Q ∈ P such that N_{S}(Q) is a Sylow p-subgroup of N_{G}(Q), there is x ∈ N_{G}(P, Q) such that xN_{S}(P)x^{−1} ≤ N_{S}(Q); and since ψ and ψ′ agree on the morphism N_{S}(P) x̂→ N_{S}(Q), they must also agree on P x̂→ Q. Thus, by taking composites, we see that for any two subgroups P, Q ∈ P, there is at least one isomorphism P x̂→ Q such that ψ_{P,Q}(x̂) = ψ′_{P,Q}(x̂).

Given an isomorphism P x̂→ Q between elements of P, any other isomorphism between these groups can be obtained by composing with an automorphism of P. It thus remains to show that ψ_{P,P} = ψ′_{P,P} for at least one subgroup in P. If the subgroups in P are p-radical, then this holds for the representative P = P_{i} (some i). Otherwise, choose P ∈ P such that N_{S}(P) is a Sylow p-subgroup of N_{G}(P), and set P = O_{p}(N_{G}(P)) ≤ S. Then for any x ∈ N_{G}(P), ψ_{P ,P} (x̂) = ψ′_{P ,P} (x̂), and hence ψ_{P,P} (x̂) = ψ′_{P,P} (x̂).

Proof of surjectivity: Now assume that no P_{i} is strictly contained in a subgroup conjugate to any P_{j} (for i, j = 1, …, m). Let L ⊆ L_{S}^{cr}(G) be the full subcategory with objects S, P_{1}, …, P_{m}. For each i,

N_{G}(P_{i}, S) = N_{G}(S)·N_{G}(P_{i}),

since if P_{i} is G-conjugate to P ≤ S, then by Alperin’s theorem again P_{i} is N_{G}(S)- conjugate to P. Hence for any (θ; θ_{1}, …, θ_{m}) ∈ X(G, G′), there is a unique functor ψ_{0} : L → L_{S′}^{cr}(G′) which sends S to S′ and P_{i} to θ(P_{i}), and whose restriction to Aut_{L}(S) or Aut_{L}(P_{i}) is θ or θ_{i}.

For each P which is G-conjugate to some P_{i}, a choice of isomorphisms P x̂→ P_{i}

and θ(P) θ̂(x) → θ(P_{i}) (for x ∈ N_{G}(S)) determines an extension of ψ_{0} to the object P. We thus get an isotypical equivalence ψ_{1} : L_{S}^{cr}(G) → L_{S′}^{cr}(G′), and this extends to an isotypical equivalence ψ : L_{S}^{c}(G) → L_{S}^{c}(G′) by Lemma 6.2.

Proposition 6.3 will now be applied to prove the second part of Theorem F. To do this, we first need to show how to construct isomorphisms N′_{G}(P) → N′_{G′}(P′) which extend a given fusion preserving isomorphism S → S′. This is the goal of the next lemma.

For a finite group G, let O_{p′}(G) denote the maximal normal subgroup of G of order prime to p. We say that G is p′-reduced if O_{p′}(G) is trivial. Notice that for any finite group G, the quotient G/O_{p′}(G) is always p′-reduced.

Lemma 6.4. Let G and G′ be finite groups, containing normal p-centric subgroups H ◁ G and H′ ◁ G′. Let S ≤ G and S′ ≤ G′ be Sylow p-subgroups. Then any fusion preserving isomorphism f : S ⟶^{≅} S′ with f(H) = H′ extends to an isomorphism

Φ : G/C′_{G}(H) ⟶^{≅} G′/C′_{G′}(H′),

and any two such extensions differ by conjugation by an element in Z(S′).

Proof. We first claim that

O_{p′}(G) = C′_{G}(H) and O_{p′}(G′) = C′_{G′}(H′).  (1)

To see this, first notice that both O_{p′}(G) and H are normal subgroups in H·O_{p′}(G) ≤ G, which is isomorphic to the direct product H × O_{p′}(G) because of the relatively prime orders. Thus O_{p′}(G) ≤ C′_{G}(H). The opposite inclusion is obvious.

Now let f : S ⟶^{≅} S′ be a fusion preserving isomorphism such that f(H) = H′. We must construct a homomorphism Φ from G/C′_{G}(H) to G′/C′_{G′}(H′), and show that it is unique up to conjugation by an element of C_{G}(S′). Upon replacing G and G′ by G/O_{p′}(G) and G′/O_{p′}(G′), respectively, and applying (1), we are reduced to the case where G and G′ are p′-reduced, and where C′_{G}(H) = C′_{G′}(H′) = 1.

We must show that f extends to an isomorphism G ^{Φ}→ G′. Consider the functors

Z : O_{S}^{c}(G) ⟶ p-Gr, and Z_{1}, Z_{2} : O_{p}(G) ⟶ p-Gr,

where Z sends each object P to Z(P), and where

Z_{1}(P) = \begin{cases}Z(P) & if P is p-centric \\ 0 & otherwise; \end{cases} and Z_{2}(P) = (Z(H))^{P} .

Then

\underleftarrow{lim}^{∗}_{O_{S}^{c}(G)}(Z) ≅ \underleftarrow{lim}^{∗}_{O_{S}(G)}(Z_{1}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G)}(Z_{1}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G)}(Z_{2}).

Here, the first isomorphism holds since C^{∗}(O_{S}^{c}(G); Z) and C^{∗}(O_{S}(G); Z_{1}) are isomorphic (no p-centric subgroup can be contained in one which is not p-centric). The second isomorphism holds since the two categories are equivalent. The third holds by [JMO, Corollary 1.8]: Z_{1}(P) ≅ Z_{2}(P) whenever P ≥ H, and otherwise N_{P H}(P)/P is a nontrivial normal p-subgroup in N_{G}(P)/P and so P is not p-radical. The functor Z_{2} is acyclic by [JMO, Proposition 5.2] (as is any functor of the form P ↦ M^{P}); and this proves that \underleftarrow{lim}^{2}_{O_{S}^{c}(G)}(Z) = 0.

By Proposition 5.4, there is an isotypical equivalence ψ : L_{S}^{c}(G) ⟶^{≅} L_{S′}^{c}(G′) such that f_{ψ} = f : S ^{≅}→ S′. Notice that Aut_{L_{S}^{c}(G)}(H) = G/C′_{G}(H) = G, since C′_{G}(H) = 1, and similarly for G′ and H′. Set

Φ = ψ_{H,H}: G = Aut_{L_{S}^{c}(G)}(H) ⟶^{≅} Aut_{L_{S′}^{c}(G′)}(H′) = G′,

regarded as an isomorphism G ^{≅}→ G′. For any g ∈ S ≤ N(H), the following square commutes

\begin{matrix}H′ & \overline{^{{ \atop Φ(g)}}→} & H′ \\ S′ & ˆ1 ; ↓ & \overline{→^{ψ(ĝ)=f̂(g)}} & S′, & ˆ1 ; ↓ \end{matrix}

since it is the image under ψ of a commutative square, which shows that Φ|_{S} = f.

It remains to prove the uniqueness of Φ up to conjugation by an element of Z(S′). Let Φ_{1}, Φ_{2} : G ^{≅}→ G′ be any two isomorphisms such that Φ_{i}|_{S} = f, and set Φ = Φ_{1}^{−1} ∘ Φ_{2} ∈ Aut(G). Thus Φ|_{S} = Id_{S}, and we must show that Φ = c_{x} for some x ∈ Z(S). Note that Φ(g) ≡ g (mod Z(H)) for all g, since for h ∈ H, g^{−1}hg = Φ(g^{−1}hg) = Φ(g)^{−1}hΦ(g).

Define γ : G/H → Z(H) by setting γ(gH) = g^{−1}Φ(g). This is a 1-cocycle whose restriction to S/H is trivial. Since the restriction map

H^{1}(G/H; Z(H)) → H^{1}(S/H; Z(H))

is injective (S/H being a Sylow p-subgroup), this implies that γ is a coboundary. In other words, there is some x ∈ Z(H) such that γ(gH) = (g^{−1}xg)x^{−1} for all g ∈ G. Thus, Φ(g) = gγ(gH) = xgx^{−1} for all g, and Φ = c_{x}. Also, x ∈ Z(S) since Φ|_{S} = Id.

We can now prove the second half of Theorem F.

Theorem 6.5. Fix a prime p and a finite group G. Assume that there is no chain P ≨ Q ≨ S, where S is a Sylow p-subgroup and P and Q are p-radical and p-centric. Then for any other finite group G′, BG_{p}^{∧} ≃ BG′_{p}^{∧} if and only if there is a fusion preserving isomorphism between Sylow p-subgroups of G and of G′.

Proof. Fix Sylow p-subgroups S ≤ G and S′ ≤ G′, and let X(G, G′) be as in Proposition 6.3. By Theorem A and Lemma 5.1, it suffices to show, for any fusion preserving isomorphism f : S → S′, that f = f_{ψ} for some isotypical equivalence ψ : L_{p}^{c}(G) → L_{p}^{c}(G′) which sends S to S′.

Let S = P_{0}, P_{1}, …, P_{m} be G-conjugacy class representatives for the p-centric pradical subgroups P ≤ S, chosen such that N_{S}(P_{i}) is a Sylow p-subgroup of N_{G}(P_{i}). For each i, P_{i} is normal and p-centric in N_{G}(P_{i}), and so Lemma 6.4 applies to show that there is an isomorphism

θ_{i} : N_{G}(P_{i})/C′_{G}(P_{i}) ^{≅}→ N_{G′}(f(P_{i}))/C′_{G′}(f(P_{i})),

whose restriction to N_{S}(P_{i}) is f. For i > 0, the restrictions of θ_{i} and θ = θ_{0} to N(P_{i}) ∩ N(S) differ by conjugation by an element of Z(N_{S}(P_{i})) by Lemma 6.4 again, and so the θ_{i} can be chosen to agree with θ on N(P_{i})∩N(S). This defines an element (θ; θ_{1}, …, θ_{m}) ∈ X(G, G′), and the result now follows from Proposition 6.3.

7. An example: BG_{2}^{∧} when G = P SL_{2}(q) and q is odd

Throughout this section, G = P SL_{2}(q), where q = p^{e} and p is an odd prime. Set k = ν_{2}(|G|), where in general 2^{ν_{2}(n)} is the highest power of 2 dividing n. Then k ≥ 2, and q ≡ 2^{k} ± 1 (mod 2^{k+1}). Fix a Sylow 2-subgroup S ≤ G, and let L_{S}^{c}(G) ⊆ L_{2}^{c}(G) denote the full subcategory whose objects are the 2-centric subgroups of G which are contained in S. As usual, this inclusion is an equivalence of categories.

For a positive integer n, C_{n} denotes a cyclic group of order n. Lemma 7.1. The Sylow 2-subgroup S in P SL_{2}(q) is dihedral of order 2^{k}. If k = 2 then N_{G}(S) ≅ A_{4}, while if k ≥ 3 then N_{G}(S) = S. If k ≥ 3, and if T_{1} and T_{2} are representatives for the two conjugacy classes of subgroups T_{i} ≅ C_{2}^{2} in S, then T_{1} and T_{2} also represent the two conjugacy classes of such subgroups in G, and N_{G}(T_{i}) ≅ Σ_{4}. Also, any subgroup of G which strictly contains N_{G}(T_{i}) is isomorphic to P SL_{2}(u), where u is such that q = u^{t} for some t.

Proof. See, for example, [Sz, Theorems 6.25–26] or [Hu, Satz 8.27] for a complete description of the subgroups of P SL_{2}(q).

Let q′ be another odd prime power and set G′ = P SL_{2}(q′). Assume ν_{2}(|G′|) = k, and let S′ ≤ G′ be a Sylow 2-subgroup (also dihedral). Let Equiv_{typ}(L_{S}^{c}(G), L_{S′}^{c}(G′)) denote, as before, the set of all isotypical equivalences modulo natural isomorphisms of functors. The next Proposition shows that the map of Proposition 6.3 is a bijection, but does not exclude the possibility that the two sets are empty. Proposition 7.2. Assume k ≥ 3. Let T_{1}, T_{2} ≤ S ≅ D_{2^{k}} be conjugacy class representatives for the subgroups isomorphic to C_{2}^{2}. Let X(G, G′) denote the set of triples (θ; θ_{1}, θ_{2}), where θ : S ⟶^{≅} S′ and the θ_{i} : N_{G}(T_{i}) ⟶^{≅} N_{G′}(θT_{i}) are all isomorphisms, and where θ_{i}|_{N_{S}(T_{i})} = θ|_{N_{S}(T_{i})} (for i = 1, 2). Then there is a bijection

Equiv_{typ}(L_{S}^{c}(G), L_{S′}^{c}(G′)) ⟶^{≅} X(G, G′)/S′.

Proof. This follows immediately from Proposition 6.3, since N_{G}(S) = S, and the only proper subgroups of S which are 2-centric and 2-radical in G are those isomorphic to C_{2}^{2}. Furthermore, C_{S}(T_{i}) ≤ S, and so (in the notation of Proposition 6.3) N′(S) = N_{G}(S) and N′(T_{i}) = N_{G}(T_{i}).

As a first consequence, we have: Corollary 7.3. Let q and q′ be odd prime powers and set G = P SL_{2}(q) and G′ = P SL_{2}(q′). Set k = ν_{2}(|G|) and k′ = ν_{2}(|G′|). Then BG_{2}^{∧} ≃ BG′_{2}^{∧} if and only if k = k′.

Proof. If BG_{2}^{∧} ≃ BG′_{2}^{∧}, then their Sylow 2-subgroups must have the same order, and hence k = k′.

Conversely, assume k = k′, let S′ be any Sylow 2-subgroup of G′, and fix an isomorphism θ : S → S′ (both are dihedral of order 2^{k} = 2^{k′}). By Theorem A, it suffices to construct an equivalence of categories L_{S}^{c}(G) → L_{S′}^{c}(G′).

If k = 2, then S ≅ C_{2}^{2} is the only object in L_{S}^{c}(G), and θ extends to an isomorphism N_{G}(S) ⟶^{≅} N_{G′}(S′) ≅ A_{4}, and hence to an equivalence of categories L_{S}^{c}(G) ⟶ L_{S′}^{c}(G′).

If k ≥ 3, then by Proposition 7.2, it suffices to show that θ extends to a triple (θ; θ_{1}, θ_{2}). Furthermore, for i = 1, 2, R_{i} =^{def} N_{S}(T_{i}) ≅ D_{8} is a Sylow 2-subgroup in N_{G}(T_{i}) ≅ Σ_{4}; and if R′_{i} = θ(R_{i}), then θ|_{R_{i}}: R_{i} → R′_{i} is fusion preserving and hence extends to an isomorphism θ_{i} : N_{G}(T_{i}) → N_{G′}(T′_{i}) by Lemma 6.4. Since S is self normalizing in G, N_{G}(S) ∩ N_{G}(T_{i}) = S ∩ N_{G}(T_{i}) = R_{i}, and thus (θ; θ_{1}, θ_{2}) ∈ X(G, G′) as defined in Proposition 7.2.

We now look at the groups

Out(BG_{2}^{∧}) ≅ Out_{typ}(L_{p}^{c}(G)) ≅ X(G, G)/S,

where S ≤ G is the Sylow 2-subgroup. The equivalence on the left is a consequence of Theorem B and the one on the right follows from Proposition 7.2. We compare those groups with the outer automorphism group Out(G). We first recall the well known description of this group.

Recall that q = p^{e} where p is prime. We regard Aut(𝔽_{q}) ≅ C_{e} as a subgroup of Aut(G): each φ ∈ Aut(𝔽_{q}) acts on G by sending a matrix A = (a_{i,j}) to the matrix φ(A) = (φ(a_{i,j})).

Lemma 7.4. Let ω ∈ Aut(G) be any automorphism induced by conjugation by an element γ ∈ P GL_{2}(q)∖P SL_{2}(q); i.e., γ = [A] for some A ∈ GL_{2}(q) whose determinant is not a square in (𝔽_{q})^{∗}. Then

Out(G) = ⟨ω⟩ × Aut(𝔽_{q}) ≅ C_{2} × C_{e}.

Proof. See [Ctr, Theorem 12.5.1]. Note that ω is called the “diagonal automorphism”, and the elements coming from Aut(𝔽_{q}) the “field automorphisms”.

We next compare the group L_{S}^{c}(G) to a certain quotient of Aut(S). The statement of the following lemma should be compared with [BL, Theorem 1.6(3)].

Lemma 7.5. Let Aut_{G}(S) denote the group of automorphisms of S induced by conjugation in G. Let

σ: Out(BG_{2}^{∧}) ≅ Out(L_{S}^{c}(G)) ⟶ Aut(S)/ Aut_{G}(S)

be the homomorphism defined by restricting an automorphism of the category to its action on Aut_{L_{S}^{c}(G)}(S) = N_{G}(S) = S.

(a) If k = 2 (so S ≅ C_{2}^{2}), then σ is an isomorphism, and Aut(S)/ Aut_{G}(S) ≅ Σ_{3}/C_{3} ≅ C_{2}.

(b) Assume k ≥ 3 (so S ≅ D_{2^{k}}), and let z ∈ S be the central element of order 2. Then σ is surjective, and

Aut(S)/ Aut_{G}(S) = Out(S) ≅ C_{2} × C_{2^{k−3}}.

Also, Ker(σ) has order 2, and under the identification of Proposition 7.2 is generated by the element [Id_{S}; Id_{T_{1}}, conj(z)_{T_{2}}].

Proof. The surjectivity of σ (for arbitrary k) follows as in the proof of Lemma 7.3, by using Lemma 6.4 to show that each θ ∈ Aut(S) extends to a triple (θ; θ_{1}, θ_{2}).

If k = 2 (if q ≡ ±3 (mod 8)), then S ≅ C_{2}^{2}, and N_{G}(S) ≅ A_{4} by Lemma 7.1. Then σ is injective since S is the only object in L_{S}^{c}(G).

If k ≥ 3, then fix generators α, β ∈ S ≅ D_{2^{k}}, where |α| = 2^{k−1} and |β| = 2. Then

Aut(D_{2^{k}}) ≅ C_{2^{k−1}} ⋊ Aut(C_{2^{k−1}}),

where the normal cyclic subgroup is generated by the automorphism α ↦ α and β ↦ αβ; and where the second factor is the subgroup of automorphisms fixing β. Also, Aut_{G}(S) = Inn(S) since N_{G}(S) = S. It now follows easily that

Aut(S)/ Aut_{G}(S) = Out(D_{2^{k}}) ≅ C_{2} × Aut(C_{2^{k−1}})/{±1} ≅ C_{2} × C_{2^{k−3}},

and it remains to calculate the kernel of σ.

Under the identification of Out_{typ}(L_{p}^{c}(G)) with X(G, G)/S given in Proposition 7.2, an element in Ker(σ) is of the form [Id_{S}; θ_{1}, θ_{2}], where θ_{i} ∈ Aut(N_{G}(T_{i})) (recall T_{i} ≅ C_{2}^{2} and N_{G}(T_{i}) ≅ Σ_{4}) are such that θ_{i}|_{N_{S}(T_{i})} = Id. One easily checks that the only nontrivial automorphism of Σ_{4} which is the identity on a subgroup D_{8} is conjugation by the central element of that subgroup. Thus, each θ_{i} is the identity or conjugation by z. And by Lemma 7.2 again, [Id_{S}, conj(z), conj(z)] = 0 in X(G, G)/S, while [Id_{S}, Id_{T_{1}}, conj(z)] ≠ 0.

By Lemma 7.4, Out(G) is a finite abelian group. We let Out(G)_{(2)} denote its Sylow 2-subgroup We are now ready to determine the group Out(BG_{2}^{∧}) and compare it to Out(G)_{(2)}.

Proposition 7.6. Fix q = p^{e}, where p is an odd prime and e ≥ 1, and set G = P SL_{2}(q). Then the natural homomorphism

B : Out(G)_{(2)} ⟶ Out(BG_{2}^{∧}) ≅ C_{2} × C_{2^{k−2}}

is always injective, and is an isomorphism if and only if p ≡ ±3 (mod 8).

Proof. We first fix Sylow 2-subgroups S ≤ G = P SL_{2}(q) and Ŝ ≤ P GL_{2}(q), such that S ≤ Ŝ has index 2, as follows. If q ≡ 1 (mod 4), then let S and Ŝ be the

subgroups generated by diagonal matrices, together with the matrix β =^{def} (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}).

If q ≡ 3 (mod 4), then identify (𝔽_{q})^{2} with 𝔽_{q^{2}}, regarded as a 2-dimensional 𝔽_{q}-vector space, and set H = ⟨(𝔽_{q^{2}})^{∗}, β⟩, where β ∈ Aut(𝔽_{q^{2}}) is the automorphism of order 2. Let Ĥ be the image of H in P GL_{2}(q), let Ŝ be the Sylow 2-subgroup of Ĥ which contains β, and set S = Ŝ ∩ P SL_{2}(q).

In either case, Ŝ ≅ D_{2^{k+1}} and S ≅ D_{2^{k}} are both dihedral groups. Fix γ ∈ Ŝ which generates the cyclic subgroup of index 2. By Lemma 7.4, conj(γ) generates the first factor in Out(G) ≅ C_{2} × Aut(𝔽_{q}).

We must compare the following maps,

\xymatrix{Out(G)_{(2)} \ar[d]_{B} \ar@{-->}[dr]^{ρ} \\ Out(BG_{2}^{∧}) \ar@{->>}[r]^{σ} & Out(S),}  (1)

where σ is surjective by Lemma 7.5. Case 1: Assume first that q ≢ 1 (mod 8). Then e is odd (q is not a square), and so Out(G)_{(2)} = ⟨ conj(γ)⟩ has order 2. Also, conj(γ) ∉ Ker(ρ) (conj(γ)|_{S} ∉ Inn(S)) by the above description of γ ∈ Ŝ. So B is injective since ρ is, and B is an isomorphism if and only if | Out(BG_{2}^{∧})| = 2, if and only if k = 2 (q ≡ ±3 (mod 8)) by Lemma 7.5. Case 2: Now assume that q ≡ 1 (mod 8). Set α = γ^{2} = [diag(v, v^{−1})], where v ∈ (𝔽_{q})^{∗} has order 2^{k}. Then S = ⟨α, β⟩ ≅ D_{2^{k}}, and conj(γ) sends α to α and β to αβ. So

Out(S) = ⟨ρ(conj(γ))⟩ × (Aut(C_{2^{k−1}})/{±1}),

where the second factor operates on ⟨α⟩ while fixing β.

If q is not a square, then | Aut(𝔽_{q})| is odd, Out(G)_{(2)} is generated by conj(γ), and so ρ (and hence B) are injective. So assume that q = (q_{1})^{2} is a square. Since q ≡ 2^{k} + 1 (mod 2^{k+1}), we have q_{1} ≡ 2^{k−1} ± 1 (mod 2^{k}). Let φ_{1} ∈ Aut(𝔽_{q}) ≤ Aut(G) denote the automorphism of order 2: φ_{1}(a) = a^{q_{1}} for a ∈ 𝔽_{q}. Since v has order 2^{k} in (𝔽_{q})^{∗}, this shows that φ_{1}(v) = −v or −v^{−1}. In particular, φ_{1}(α) = α or α^{−1} (where α = diag(v, v^{−1})), so φ_{1}|_{S} ∈ Inn(S), and φ_{1} ∈ Ker(ρ). Furthermore, Ker(ρ) is generated by φ_{1}, since any automorphism φ_{2} ∈ Aut(𝔽_{q}) of order 4 is of the form (a ↦ a^{q_{2}}) where q_{2} ≡ 2^{k−2} ± 1 (mod 2^{k−1}).

Define Ψ ∈ Aut(G), by setting

Ψ = \begin{cases}φ_{1} & if q_{1} ≡ 1 (mod 2^{k−1}) \\ φ_{1} ∘ conj(β) & if q_{1} ≡ −1 (mod 2^{k−1}). \end{cases}

These are chosen so that [Ψ] = [φ_{1}] in Out(G), and Ψ|_{S} = Id. Set z = diag(i, −i) = α^{2^{k−2}}, the central element of order two in S, and set T_{1} = ⟨z, β⟩ and T_{2} = ⟨z, αβ⟩. By Lemma 7.1, T_{1} and T_{2} represent the two G-conjugacy classes of subgroups isomorphic to C_{2}^{2}.

Since Ψ|_{T_{1}} = Id, Ψ|_{N_{G}(T_{1})} must be conjugation by some element of T_{1} (just look at the automorphisms of Σ_{4}). Furthermore, Ψ is the identity on N_{S}(T_{1}) ≅ D_{8}, and so Ψ|_{N_{G}(T_{1})} must be conjugation by an element of Z(N_{S}(T_{1})) = {1, z}. After replacing Ψ by Ψ ∘ conj(z) if necessary, we can assume that Ψ|_{N_{G}(T_{1})} = Id. The same argument shows that Ψ|_{N_{G}(T_{2})} is conjugation by 1 or z. By Lemma 7.1, G′ =^{def} ⟨N_{G}(T_{1}), N_{G}(T_{2})⟩ ≅ P GL_{2}(q′) for some q′ such that q = (q′)^{t}, and t must be odd since S ≤ G′. Then Ψ|_{G′} ≠ Id, since restriction induces an isomorphism Out(G)_{(2)} ≅ Out(G′)_{(2)} by Lemma 7.4. Thus Ψ|_{N_{G}(T_{2})} ≠ Id, and hence Ψ|_{N_{G}(T_{2})} = conj(z). So by Lemma 7.5, B(Ψ) is the nontrivial element in Ker[Out(BG_{2}^{∧}) ^{σ}→ Out(S)], and thus B is injective.

To see when B is an isomorphism, note first that

| Out(BG_{2}^{∧})| = 2·| Out(S)| = 2^{k−1} and | Out(G)_{(2)}| = 2^{m+1} (m = ν_{2}(e))

by Lemmas 7.4 and 7.5. So B is an isomorphism if and only if p has order 2^{k−1} = 2^{m+1} in (ℤ/2^{k+1})^{∗}; and this is the case if and only if p ≡ ±3 (mod 8). In particular, since BG_{2}^{∧} ≃ BP SL_{2}(5^{2^{k−2}})_{2}^{∧} by Proposition 7.3,

Out(BG_{2}^{∧}) ≅ Out(BP SL_{2}(5^{2^{k−2}})_{2}^{∧}) ≅ Out(P SL_{2}(5^{2^{k−2}})) ≅ C_{2} × C_{2^{k−2}}.

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56 Homotopy equivalences of p-completed classifying spaces

Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

E-mail address: broto@mat.uab.es

Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K.

E-mail address: ran@maths.abdn.ac.uk

LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

E-mail address: bob@math.univ-paris13.fr